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? ; ■I 1 > 11 ■ reference collection book KC Kansas city public library kansas city, missouri iP^ ^U' Ns^ Sii/ From the collection of the n R m o Jrre linger V * library p San Francisco, California 2008 f • « fl • • • • • « • ■ • HE BEL Li:iii8L^^S:'^T-,,E\M p.', j'i'— ' , meat journal l^^r/ A IN ^OTED TO THE SCIENTIFIC ^^r>^ AND ENGINEERING »ECTS OF ELECTRICAL COMMUNICATION U M E XXXV JANUARY 1956 tf k k--- • ' t. N U M B E R-lv DiflPused Emitter and Base Silicon Transistors J ^' ^ '^ ^ ^^^° M. TANENBAUM AND D. E. THOMAS 1 A High-Frequency Diffused Base Germanium Transistor c. a. lee 23 Waveguide Investigations with Millimicrosecond Pulses a. c. beck 35 Experiments on the Regeneration of Binary Microwave Pulses o. B. delange 67 Crossbar Tandem as a Long Distance Switching System a. O. ADAM 91 Growing Waves Due to Transverse Velocities J. R. pierce and l. r. walker 109 Coupled Helices j. s. cook, r. kompfner and c. f. quatb 127 Statistical Techniques for Reducing the Experiment Time in Re- liability Studies MILTON sobel 179 A Class of Binary Signaling Alphabets david slepian 203 Bell System Technical Papers Not Published in This Journal 235 Recent Bell System Monographs 242 Contributors to This Issue 244 COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY ; , * -^ -^ f - -.r » ' J " -' • THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. E. K A P P E L, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Executive Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. MCMILLAN, Chairman H. R. HUNTLEY A. J. BUSCH F. R. LACK A. C. DICKIESON J. R. PIERCE R. L. DIETZOLD H. V. SCHMIDT K. E. GOULD C. E. SCHOOLEY E. L GREEN G. N. THAYER EDITORIAL STAFF J. D. TEBO, Editor M. E. s T R I E B Y, Managing Editor R. L. SHEPHERD, Production Editor THE" BELL SYSTEM TECHNICAL JOURNAL is pubUshed six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV JANUARY 1956 number 1 Copyright 1956, American Telephone and Telegraph Company Diffused Emitter and Base Silicon Transistors* By M. TANENBAUM and D. E. THOMAS (Manuscript received October 21, 1955) Silicon n-p-n transistors have been made in which the base and emitter regions were produced by diffusing impurities from the vapor phase. Tran- sistors with base layers 3.8 X 10~ -cm thick have been made. The diffusion techniques and the processes for making electrical contact to the structures are described. The electrical characteristics of a transistor with a maximum alpha of 0.97 and an alpha-cutoff of 120 mc/sec are presented. The manner in which some of the electrical parameters are determined by the distribution of the doping impurities is discussed. Design data for the diffused emitter, dif- fused base structure is calcidated and compared with the rneasured char- acteristics. INTRODUCTION The necessity of thin base layers for high-frequency operation of tran- sistors has long been apparent. One of the most appealing techniques for controlling the distribution of impurities in a semiconductor is the dif- fusion of the impurity into the solid semiconductor. The diffusion co- efficients of Group III acceptors and Group V donors into germanium and silicon are sufficiently low at judiciously selected temperatures so * A portion of the material of this paper was presented at the Semiconductor Device Conference of the Institute of Radio Engineers, Philadelphia, Pa., June, 1955. 2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 that it is possible to envision transistors with base layer thicknesses of a micron and frequency response of several thousand megacycles per second. A major deterent to the application of diffusion to silicon transistor fabrication in the past was the drastic decrease in lifetime which generally occurs when silicon is heated to the high temperatures required for dif- fusion. There was also insufficient knowledge of the diffusion parameters to permit the preparation of structures with controlled layer thicknesses and desired dopings. Recently the investigations of C. S. Fuller and co- workers have produced detailed information concerning the diffusion of Group III and Group V elements in silicon. This information has made possible the controlled fabrication of transistors with base layers suffi- ciently thin that high alphas are obtained even though the lifetime has been reduced to a fraction of a microsecond. In a cooperative program with Fuller, diffusion structures were produced which have permitted the fabrication of transistors whose electrical behavior closely approxi- mates the behavior anticipated from the design. This paper describes these techniques which have resulted in high alpha silicon transistors with alpha-cutoff of over 100 mc/sec. 1.0 FABRICATION OF THE TRANSISTORS Fuller's work has shown that in silicon the diffusion coefficient of a Group III acceptor is usually 10 to 100 times larger than that of the Group V donor in the same row in the periodic table at the same tem- peratures. These experiments were performed in evacuated silica tubes using the Group III and Group V elements as the source of diffusant. Under these conditions a particular steady state surface concentration of the diffusant is produced and the depth of diffusion is sensitive to this concentration as well as to the diffusion coefficient. The experiments show that the effective steady state surface concentration of the donor impurities produced under these conditions is ten to one hundred times greater than that of the acceptor impurities. Thus, by the simultaneous diffusion of selected donor and acceptor impurities into n-type silicon an n-p-n structure will result. The first n-la,yer forms because the surface concentration of the donor is greater than that of the acceptor. The p-laycr is protluced because the acceptor diffuses faster than the donor and gets ahead of it. The final n-region is simply the original background doping of the n-type silicon sample. It has been possible to produce n-p-n structures by the simultaneous diffusion of several combinations of donors and acceptors. Often, however, the diffusion coefficients and surface concentrations of the donors and acceptors are such that opti- 1 C. S. Fuller, private communication. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 3 mum layer thicknesses (see Sections 3 and 4) are not produced by simul- taneous diffusion. In this case, one of the impurities is started ahead of the other in a prior diffusion, and then the other impurity is diffused in a second operation. With the proper choice of diffusion temperatures and times it has been possible to make n-p-n structures with base layer thicknesses of 2 X 10~* cm. The uniformity of the layers in a given specimen is better than ten per cent of the layer thickness. Fig. 1 illustrates the uniformity of the layers. This figure is an enlarged photograph of a view perpendicular to the surface of the specimen. A bevel which makes an angle of five degrees with the original surface has been polished on the specimen. This angle magnifies the layer thickness by 11.5. The layer is defined by an etchant which preferentially stains p-type silicon^ and the width of the layer is measured with a calibrated microscope. After diffusion the entire surface of the silicon wafer is covered with the diffused n- and p-type layers, see Fig. 2(a). Electrical contact must now be made to the three regions of the device. The base contact can be made by polishing a bevel on the specimen to expose and magnify the base layer and then alloying a lead to this region by the same tech- f.^ *f^'- *; '>i i * /i n-TfPE DIFFUSED LAV^ER fo-t^^E*OiFFUSED LAYER i»# OF^GIt^L n-TYPE CRYSTAl. I 1 EQUIVALENT TO 2 X lO"'* CM LAYER THICKNESS Fig- 1 — Angle section of a double diffused silicon wafer. The p-type center ayer is approximately 2 X 10-< cm thick. 4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 niques employed in the fabrication of grown junction transistors. Fig. 2(b). However, a much simpler technique has been evolved. If the sur- face concentration of the donor diffusant is maintained below a certain critical value, it is possible to alloy an aluminum wire directly through the diffused n-type layer and thus make effective contact to the base layer, Fig. 2(c). Since the resistivity of the original silicon wafer is one to five ohm-cm, the aluminum will be rectifying to this region. It has been experimentally shown that if the surface concentration of the donor diffusant is less than the critical value mentioned above, the aluminum will also be rectifying to the diffused n-type region and the contact becomes merely an extension of the base layer. The n-layers produced by diffusing from elemental antimony are below the critical concentration and the direct aluminum alloying technique is feasible. -n + TYPE DIFFUSED LAYER -p-TYPE DIFFUSED LAYER n + n+ -ALUMINUM WIRE p + ALUMINUM DOPED REGROWTH LAYER n-TYPE (b) ,^- ALUMINUM WIRE P + ALUMINUM DOPED , REGROWTH LAYER ^M'nY ^-i-r n-TYPE (c) Fig. 2 — ■ Schematic illustralioii of (a) double diffused n-p-n wafer, (b) angle section method of making base contact, and (c) direct alloying method of making base contact. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS AU-Sb PLATED POINT VAPORIZED Al LINE 0.005 CM WIDE t MM Fig. 3 — Mounted double diffused transistor. Contact to the emitter layer is achieved by alloying a film of gold containing a small amount of antimony. Since this alloy will produce an n-type regrowth layer, it is only necessary to insure that the gold- antimony film does not alloy through the p-type base layer, thus shorting the emitter to the collector. This is controlled by limiting the amount of gold-antimony alloy which is available by using a thin evaporated film or by electroplating a thin film of gold-antimony alloy on an inert metal point and alloying this structure to the emitter layer. Ohmic, contact to the collector is produced by alloying the silicon wafer to an inert metal tab plated with a gold-antimony alloy. 6 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The transistors whose characteristics are reported in this paper were prepared from 3 ohm-cm n-type siHcon using antimony and ahmiinum as the diffusants. The base contact was produced by evaporating alumi- num through a mask so that a hne approximately 0.005 X 0.015 cm in o lateral dimensions and 100,000 A thick was formed on the surface. This aluminum line was alloyed through the emitter layer in a subsequent operation. The wafer was then alloyed onto the plated kovar tab. A small area approximately 0.015 cm in diameter was masked around the line and the wafer was etched to remove the unwanted layers. The unit was then mounted in a header. Electrical contact to the collector was made by soldering to the kovar tab. Contact to the base was made with a tungsten point pressure contact to the alloyed aluminum. Contact to the emitter was made by bringing a gold-antimony plated tungsten point into pressure contact with the emitter layer. The gold-antimony plate was then alloyed by passing a controlled electrical pulse between the plated point and the transistor collector lead. Fig. 3 is a photograph of a mounted unit. 2.0 ELECTRICAL CHARACTERISTICS The frequency cutoffs of experimental double diffused silicon tran- sistors fabricated as described above are an order of magnitude higher than the known cutoff frequencies of earlier silicon transistors. This is shown in Fig. 4 which gives the measured common base and common emitter current gains for one of these units as a function of frequency. The common base short-circuit current gain is seen to have a cutoff fre- quency of about 120 mc/sec. The common emitter short-circuit current gain is shown on the same figure. The low-freciuency current gain is better than thirty decibels and the cutoff frequency which is indicated by the freciuency at which the gain is 3 db below its low-frequency value is 3 mc/sec. This is an exceptionally large common emitter band- width for a thirty db common emitter current gain and is of the same order of magnitude as that obtained with the highest frequency ger- manium transistors (e.q., p-n-i-p or tetrode) which had been made prior to the diffused base germanium transistor. ^ Tlio iiicroasp in (•oiiiinon haso current gain ahovc unity (indicated by current gain in decibels being positive) in the vicinity of 50 mc/sec is caused by a reactance gain error in the common base measurement. This error is caused by a combination of the emitter to ground parasitic capacitance and the i)ositive reactance com- ponent of the transistor input impedance resulting from phase shift in the ali)ha current gain. ' C. A. Lee, A High-Frequency Diffused Base Germanium Transistor, see page 23. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS z < o I- z LJ a. cr D O 40 30 20 (0 -\0 -20 -30 Ie = 3 MA Vc = 10 VOLTS COMMON^ EMITTER N 'OCCB — ^ ^^ OCq = 0.9716 ['=^"=106MC l-Ofg \ facb = i20MC \ COMMON BASE \ \ \ 0.1 0.2 0.5 1.0 2 6 10 20 50 100 200 FREQUENCY IN MEGACYCLES PER SECOND 500 1000 Fig. 4 — ■ Short-circuit current gain of a double diffused silicon n-p-n transistor as a function of frequency in the common emitter and common base connections. Fig. 5 shows a high-freciueiicy lumped constant equivalent circuit for the double diffused silicon transistor whose current gain cutoff char- acteristic is shown in Fig. 4. External parasitic capacitances have been omitted from the circuit. The configuration is the conventional one for junction transistors with two exceptions. A series resistance rj has been added in the emitter circuit to account for contact resistance resulting from the fact that the present emitter point contacts are not perfectly ohmic. A second resistance r/ has been added in the collector circuit to account for the ohmic resistance of the n-type silicon between the col- lector terminal and the effective collector junction. This resistance exists in all junction transistors but in larger area low frequency junction transistors its effect on alpha-cutoff is sufficiently small so that it has been ignored in equivalent circuits of these devices. The collector RC Ce = TmmF Pq -]AU) Cc = 0.52//^F r ' _ ,50 co Tg = 150; a J^C( •Le '%=QOCO COMMON BASE CURRENT GAIN CUT-OFF FREQUENCY ■ 120 MC Ic = 3 MA Vc = 10 VOLTS Fig. 5 ~ High-frequency lumped constant equivalent circuit for a double diffused silicon n-p-n transistor. 8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 cutoff caused by the collector capacitance and the combined collector body resistance and base resistance is an order of magnitude higher than the measured alpha cutoff frequency and therefore is not too serious in impairing the very high-frecjuency performance of the transistor. This is due to the low capacitance of the collector junction which is seen to be approximately 0.5 mmf at 10 volts collector voltage. The base resistance of this transistor is less than 100 ohms which is quite low and compares very favorably with the best low frequency transistors reported previously. The low-frequency characteristics of the double diffused silicon tran- sistor are very similar to those of other junction transistors. This is il- lustrated in Fig. 6 where the static collector characteristics of one of these transistors are given. At zero emitter current the collector current is too small to be seen on the scale of this figure. The collector current 45 40 35 30 25 20 15 10 -5 le=0 2 4 6 8 10 12 ] J 14/ ^ J^ ^ y^ ^ 2 4 6 8 10 12 14 CURRENT, If, IN MILUAMPERES Fig. 6 — Collector characteristics of a double diffused silicon n-p-n tran- sistor. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 9 0.98 0.94 0.90 0.86 a 0.82 0.78 0.74 0.70 T=150°C, ^ ^ ^ ^ ^ 7 <^ y ^ ^ ^ \ / 9/ y 24, 5M 65-W /> 7 /24.5 t35^y\ 7 15ol / / 1 1 1 _L. 1 1 1 ,1 0.1 0.2 0.4 0.6 1 2 4 6 8 10 20 CURRENT, Ig, IN MILLIAMPERES Fig. 7 — Alpha as a function of emitter current and temperature for a double diffused silicon n-p-n transistor. under this condition does not truly saturate but collector junction re- sistance is very high. Collector junction resistances of 50 megohms at reverse biases of 50 volts are common. The continuous power dissipation permissible with these units is also shown in Fig. 6. The figure shows dissipation of 200 milliwatts and the units have been operated at 400 milliwatts without damage. As illus- trated in Fig. 3 no special provision has been made for power dissipation and it would appear from the performance obtained to date that powers of a few watts could be handled by these iniits with relatively minor provisions for heat dissipation. However, it can also be seen from Fig. 6 that at low collector voltages alpha decreases rapidly as the emitter current is increased. The transistor is, therefore, non-linear in this range of emitter currents and collector voltages. In many applications, this non-linearity may limit the operating range of the device to values below those which would be permissible from the point of view of con- tinuous power dissipation. Fig. 7 gives the magnitude of alpha as a function of emitter current for a fixed collector voltage of 10 volts and a number of ambient tem- peratures. These curves are presented to illustrate the stability of the parameters of the double diffused silicon transistor at increased ambient temperatures. Over the range from 1 to 15 milliamperes emitter current and 25°C to 150°C ambient temperature, alpha is seen to change only 10 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 by approximately 2 per cent. This amounts to only 150 parts per million change in alpha per degree centigrade change in ambient temperature. The decrease in alpha at low emitter currents shown in Fig. 7 has been observed in every double diffused silicon transistor which has been made to date. Although this effect is not completely understood at present it could be caused by recombination centers in the base layer that can be saturated at high injection levels. Such saturation would result in an increase in effective lifetime and a corresponding increase in alpha. The large increase in alpha with temperature at low emitter currents is con- sistent with this proposal. It has also been observed that shining a strong light on the transistor will produce an appreciable increase in alpha at low emitter currents but has little effect at high emitter currents. A strong light would also be expected to saturate recombination centers which are active at low emitter currents and this behavior is also con- sistent with the above proposal. 3.0 DISCUSSION OF THE TRANSISTOR STRUCTURE Although the low frequency electrical characteristics of the double diffused silicon transistor which are presented in Section 2 are quite similar to those usually obtained in junction transistors, the structure of the double diffused transistor is sufficiently different from that of the grown junction or alloy transistor that a discussion of some design principles is warranted. This section is devoted to a general discussion of the factors which determine the electrical characteristics of the tran- sistors. In Section 4 the general ideas of Section 3 are applied in a more specialized fashion to the double diffused structure and a detailed cal- culation of electrical parameters is presented. One essential difference between the double diffused transistor and grown junction or alloy transistors arises from the manner in which the impurities are distributed in the three active regions. In the ideal case of a double-doped grown junction transistor or an alloy transistor the concentration of impurities in a given region is essentially uniform and the transition from one conductivity type to another at the emitter and collector junctions is abrupt giving rise to step junctions. On the other hand in the double diffused structure the distribution of impurities is more closely described by the error function complement and the emitter and collector junctions are graded. Tlu\se differences can have an appre- ciable influence on the electrical beha\'ior of the transistors. Fig. 8(a) shows the probable distribution of donor impurities, No , and acceptor impurities, A''^ , in a double diffused n-p-n. Fig. 8(b) is a DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 11 DONORS ACCEPTORS DISTANCE (a) DISTANCE *• (b) Fig. 8 — Diagrammatic representation of (a) donor and acceptor distributions and (b) uncompensated impuritj- distribution in a double diffused n-p-n tran- sistor. plot of Nd — Na which would result from the distribution in Fig. 8(a). Kromer has shown that a nonuniform distribution of impurities in a semiconductor will produce electric fields which can influence the flow of electrons and holes. For example, in the base region the fields between the emitter junction, Xe , and the minimum in the Nd — Na curve, x', will retard the flow of electrons toward the collector while the fields between this minimum and the collector jvmction, Xc , will accelerate the flow of electrons toward the collector. These base laj^er fields will affect the transit time of minority carriers across the base and thus contribute * H. Kromer, On Diffusion and Drift Transistor Theory I, II, III, Archiv. der Electr. Ubertragung, 8, pp. 223-228, pp. 363-369, pp. 499-504, 1954. 12 THE BELL SYSTEM TECHNICAL JOUENAL, JANUARY 1956 to the fre(iuency response of the transistor. In addition the base re- sistance will be dependent on the distribution of both diffusants. These three factors are discussed in detail below. Moll and Ross have determined that the minority current, /,„ , that will flow into the base region of a transistor if the base is doped in a non- uniform manner is given by f N(x) dx where rii is the carrier concentration in intrinsic material, q is the elec- tronic charge, V is the applied voltage, Dm is the diffusion coefficient of the minority carriers, and the integral represents the total number of uncompensated impurities in the base. The primary assumptions in this derivation are (1) planar junctions, (2) no recombination in the base region, and (3) a boundary condition at the collector junction that the minority carrier density at this point equals zero. It is also assumed that the minority carrier concentration in the base region just adjacent to the emitter junction is equal to the equilibrium minority carrier density at this point multiplied by the Boltzman factor exp (qV/kT). It is of special interest to note that Im depends only on the total number of uncom- pensated impurities in the base and not on the manner in which they are distributed. In the double diffused transistor, it has been convenient from the point of ease of fabrication to make the emitter layer approximately the same thickness as the base layer. It has been observed that heating sili- con to high temperatures degrades the lifetime of n- and p-type silicon in a similar manner. Both base and emitter layers have experienced the same heat treatment and to a first approximation it can be assumed that the lifetime in the two regions will be essentially the same. Thus as- sumptions (1) and (2) should also apply to current flow from base to emitter. If we assume that the surface recombination \'elocity at the free surface of the emitter is infinite, then this imposes a boundary condition at this side of the emitter which under conditions of forward bias on the emitter is equivalent to assumption (3). Thus an equation of the form of (3.1) should also give the minority current flow from base to emitter. Since the emitter efficiency, y, is given by ^ J. Tj. Moll and I. M. Ross, The J)opendencc of Transistor Paramotors on tlie Distribution of Base Layer liesistivity, Proc. I.R.E. in press. 8 G. Bemski, private comnmnication. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 13 /m (emitter to base) -y = . . . /^(emitter to base) + /„j(base to emitter) proper substitution of (3.1) will give the emitter efficiency of the double diffused n-p-n transistor, 1 7 = J-'n Z).^''^-^"^ dx p .6 (3.2) \ (No - iVj dx In (3.2), Dp is the diffusion coefficient of holes in the emitter, /)„ is the diffusion coefficient of electrons in the base and the ratio of integrals is the ratio of total uncompensated doping in the base to that in the emitter. A calculation of transit time is more difficult. Kromer has studied the case of an aiding field which reduces transit time of minority carriers across the base region and thus increases frequency response. In the double diffused transistor the situation is more complex. Near the emitter side of the base region the field is retarding (Region R, see Fig. 8) and becomes aiding (Region A) only after the base region doping reaches a maximum. The case of retarding fields has been studied by Lee and by MoU.^ At present, the case for a base region containing both types of fields has not been solved. However, at the present state of knowledge some speculations about transit time can be made. The two factors of primary importance are the magnitude of the built-in fields and the distance over which they extend. In the double diffused transistor, the widths of regions R and A are determined by the surface concentrations and diffusion coefficients of the diffusants. It Can be shown by numerical computation that if region R constitutes no more than 30-40 per cent of the entire base layer width, then the overall effect of the built-in fields will be to aid the transport of minority car- riers and to produce a reduction in transit time. In addition the absolute magnitude of region R is important. If the point x' should occur within an effective Debye length from the emitter junction, i.e., if x' is located in the space charge region associated with the emitter junction, then the retarding fields can be neglected. The base resistance can also be calculated from surface concentrations and diffusion coefficients of the impurities. This is done by considering the base layer as a conducting sheet and determining the sheet con- ' J. L. Moll, private communication. 14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 ductivity from the total number of uncompensated impurities per square centimeter of sheet and the approjiriate moliility weighted to account for impurity scattering. 4.0 CALCULATION OF DESIGN PARAMETERS To calculate the parameters which determine emitter efficiency, transit time, and base resistance it is assumed that the distribution of uncom- pensated impurities is given by N(x) = Nicrfc f - N-2erJc^ + Nz (4.1) where A^i and A^2 are the surface concentrations of the emitter and base impurity diffusants respectively, Li and L^ are their respective diffusion lengths, and Nz is the original doping of the semiconductor into which the impurities are diffused. The impurity diffusion lengths are defined as Li = 2 V/M and L2 = 2 ^Ddo (4.2) where the D's are the respective diffusion coefficients and the f's are the diffusion times. Equation (4.1) can be reduced to r(^) = Ti erfc I - Ta erfc X^ + 1 (4.3) where For cases of interest here, r(^) will be zero at two points, a and 13, and will have one minimum at ^'. In the transistor structure the emitter junction occurs at ^ = ^v and the collector junction occurs at ^ = (3. Thus the base width is determined by 13 — a. The extent of aiding and retarding fields in the base is determined by ^'. The integral of (4.3) from to a, I\ , and from o to ^, I2 , are the integrals of interest in (3.2) and thus determine emitter efficiency. In addition I2 is the integral from which base resistance can be calculated. The calculations which follow apply only for values of ri/r2 and To greater than ten. Some of the simplifying assumptions which are made will not apply at lower values of these parameters where the distribution of both diffusants as well as the background doping affect the structure in all three regions of the device. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 15 4.1 Base Width From Fig. 8 and (4.3) it can be seen that for r2 ^ 10, a is essentially independent of r2 and is primarily a function of T1/T2 and X. Fig. 9 is a plot of a versus ri/r2 with X as the parameter. The particular plot is for r2 = 10 . Although as stated a is essentially independent of r2 , at lower values of r2, a may not exist for the larger values of X, i.e., the p-layer does not form. In the same manner, it can be seen that ^ is essentially independent of T]/T2 and is a function only of r2 and X. Fig. 10 is a plot of /3 versus F^ with X as a parameter. This plot is for Ti/Fo = 10 and at larger Fi/Fo , /3 may not exist at large X. 10" \0' 10 r2=)o'' /// // / ^ ::i ll r / / m 0/ / ' > /os/ 1 i 1 /// 'o.e/ / f 0.7/ / /// / / <.e I w. W / / / 1.0 1.4 1.8 2.2 2.6 a 3.0 3.4 3.8 Fig. 9 — Emitter layer thickness (in reduced units) as a function of the ratio of the surface concentrations of the diffusing impurities (ri/r2) and the ratio of their diffusion lengths (X). 16 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The base width W = ^ — a can be obtained from Figs. 9 and 10. a, 13 and iv can be converted to centimeters by nuiltiplying by the appropriate value of Li . 4.2 Emitter Efficiency With the hmits a and /3 determined above, the integrals h and 1 2 can be calculated. Examination of the integrals shows that h is closely pro- portional to ri/r2 and also to r2 . On the other hand I2 is closely propor- tional to r2 and essentially independent of ri/r2 . Thus, the ratio of /2//1 which determines 7 depends primarily on ri/r2 . Fig. 11 is a plot of the constant /2//1 contours in the ri/T2 — X plane for lo/h ii^ the range from — 1.0 to —0.01. The graph is for r2 = 10 . Since from (3.2) 7 = 1 1 _ ^h Dnh (4.4) for an n-p-n transistor, and assuming Dp/Dn = /^ for silicon, then to' (0- 10' 10 1' 1 \= ..J\ 0.6- 0.5- ::ffl M \u |6 In 1 1° 1 \\\ ( 0.2 0.1 '/// /// 0.01/ ill 7 / / /// / / / 10 20 50 100 200 500 1000 Fig. 10 — (Collector junction dopth (in rodurod units) as a function of the sur- face concuMit.ration (in reduced units) of llie dilfusaiit wliicli determines the con- ductivity type of the l)ase layer (I'.') and liie ratio of tlie dilTusioii lengths (X) of the tAvo diffusing inii)urifies. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 10" 17 10 H Ta 10 10 r2 = io'* 2 w \v V 2 ? 1 ^ 1 \\ \ t ^. \ \I2/I 1 2 V ,\ N-0 VO.05 02 -i.o\ -0.3S^ 32X^0 '\ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 11 — ^Dependence of emitter efficiency upon diffusant surface concentra- tions and diffusion lengths. The lines of constant /2//1 are essentially lines of constant emitter efficiency. The ordinate is the ratio of surface concentrations of the two diffusants and the abscissa is the ratio of their diffusion lengths. /2//1 = — 1.0 corresponds to a 7 of 0.75 and /2//1 = —0.01 corresponds to a 7 of 0.997. 4.. 3 Base Resistance It was indicated above that I2 depends principally on r2 and X. Fig. 12 is a plot of the constant I2 contours in the r2 — X plane for I2 in the range from —10^ to —10. The graph is for Ti/To = 10. The base layer sheet conductivity, cjb , can be calculated from these data as Qb = —qtihTjiNz (4.5) where q, L\ and A^3 are as defined above and /I is a mobility properly weighted to account for impurity scattering in the non-uniformly doped base region. The units of gb are mhos per square. 18 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 10- 1 2= -10,00^ / / 7/ 1 / -5000/ r / / // / / / -1000/ // // / / 1 2 1 / /-5oa / / / / / 1 ^/^^ / / / /I ^/ / / 1 1 10 / / // v. /-ioy V 11 / / / /, // /-/ , (I 5 // /, -^ /J / V/ / 2 ^ ^ ^ f^ u 10 102 r / / ^ / / ^ 5 — 1 ^/ V r 10 / / 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 12 — Dependence of base layer sheet condiictivitj^ on diffusant surface concentrations and diffusion lengths. The lines of constant Ii are essentiallj' lines of constant base sheet conductivity. The ordinate is the surface concentration (in reduced units) of the diffusant which determines the conductivity type of the base layer and the abscissa is the ratio of the diffusion lengths of the two difi'using impurities. 4.4 Transit Time With a knowledge of where the minimum value, ^', of (4.3) occurs, it is possible to calculate over what fraction of the base width the fields are retarding. The interesting quantity here is 13 - a ^ is a function of ri/r2 and X and varies only very slowly with ri/r2 . a is also a function of ri/r2 and X and varies only slowly with ri/r2 . The most rapidly changing part of bJi is l^ which depends primarily on r2 as noted above. Fig. 13 is a plot of the constant LR contours in the r2 — X plane for values of A/2 in the range 0.1 to 0.3. This graph is DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 19 lor data with ri/r2 = 10. As ri/r2 increases at constant r2 and X, AR decreases slightly. At ri/r2 = 10\ the average change in AR is a decrease of about 25 per cent for constant r2 and X when AR ^ 0.3. The error is larger for values of AR greater than 0.3. It was noted above that when AR becomes greater than 0.3, the retarding fields become dominant. Therefore, this region is of slight interest in the design of a high frequency transistor. 4.5 A Sample Design By superimposing Figs. 11, 12 and 13 the ranges of r2 , ri/r2 and X which are consistent with desired values of y, gt and AR can be deter- 0.7 Fig. 1.3 — Dependence of the built-in field distribution on concentrations and diffusion lengths. The lines of constant aR indicate the fraction of the base layer thickness over which built-in fields are retarding. The ordinate is the surface concentration (in reduced units) of the diffusant which determines the conductiv- ity type of the base layer and the abscissa is the ratio of the diffusion lengths of the two diffusing impurities. 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 mined by the area enclosed by the specified contour lines. It is also possible to compare the measured parameters of a specific device and observe how closely they agree with what is predicted from the estimated concentrations and diffusion coefficients. This is done below for the transistor described in Sections 1 and 2. The comparison is complicated by the fact that the exact values of the surface concentrations and diffusion coefficients are not known {Precisely enough at present to permit an accurate evaluation of the design theory. However, the following values of concentrations and diffusion coefficients are thought to be realistic for this transistor. iVi = 5 X 10^' /)i = 3 X 10"'' /i = 5.7 X lO' iV2 = 4 X 10'' Di = 2.5 X 10"" t^= 1.2 X lO' Nz = 10'' From these values it is seen that Ti/ra = 12.5; r, = 400; X = 0.6 From Fig. 9, a = 1.9 and from Fig. 10, /3 = 3.6 and therefore w = 1.7. Measurement of the emitter and base layer dimensions showed that these layers were approximately the same thickness which was 3.8 X 10" cm. Thus the ifieasured ratio of emitter width to base width of unity is in good agreement with the ^'alue of 1.1 predicted from the assumed con- centrations and diffusion coefficients. From Fig. 11, lo/h ~ —0.01. If this value is substituted into (4.4), 7 = 0.997. This compares with a measured maximum alpha of 0.972. From Fig. 12, lo = —15. Assuming an average hole mobility of 350 cm' /volt. sec. and evaluating Li from the measured emitter thickness and the calculated a, (4.5) gives a value of gb = 1.7 X 10^ mhos per square. The geometry of the emitter and base contacts as shown in Fig. 3 makes it difficult to calculate the effective base resistance from the sheet conductivity even at very small emitter currents. In addition at the very high inje{;tion levels at which these transistors are operated the calculation of effective base resistance becomes very difficult. However, from the geometr}^ it would be expected that the effective base re- sistance would l)c no greater than 0.1 of the sheet resistivity or 600 ohms. This is about seven times larger than the measured \'alue of 80 ohms reported in Section 2. From Fig. b3, A/^ is approximately 0.20. Thus there should be an over- all aiding elfect of the built-in fields. In addition the impurity gradient at the emitter junction is believed to be approximately lO'Vcm and the DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 21 space charge associated with this gradient will extend approximately 2 X 10 ■' cm into the base region. The base thickness over which re- tarding fields extend is AR times the base width or 7.6 X 10~^ cm. Thus the first quarter of region R will be space charge and can be neglected. The frequency cutoff from pure diffusion transit is given by 2A3D ,. , where W is the measured base layer thickness. Assuming D — 25 cmVsec for electrons in the base region, ,/'„ = (w mc/sec. Since the measured cutoff was 120 mc/sec, the predicted aiding effect of the built-in field is evidently present. These computations illustrate how the measured electrical parameters can be used to check the values of the surface concentrations and dif- fusion coefficients. Conversely knowledge of the concentrations and diffusion coefficients aid in the design of devices which will have pre- scribed electrical parameters. The agreement in the case of the transistor described above is not perfect and indicates errors in the proposed values of the concentrations and diffusion coefficients. However, it is sufficiently close to be encouraging and indicate the value of the calculations. The discussion of design has been limited to a very few of the important parameters. Junction capacitances, emitter and collector resistances are among the other important characteristics which have been omitted here. Presumably all of these quantities can be calculated if the detailed structure of the device is known and the structure should be susceptible to the type of analysis used above. Another fact, which has been ignored, is that these transistors were operated at high injection levels and a low level analysis of electrical parameters was used. All of these other factors must be considered for a detailed understanding of the device. The object of this last section has been to indicate one path which the more detailed analysis might take. 5.0 CONCLUSIONS By means of multiple diffusion, it has been possible to produce silicon transistors with alpha-cutoff above 100 mc/sec. Refinements of the described technicjues offer the possibility of even higher frequency per- formance. These transistors show the other advantages expected from silicon such as low saturation currents and satisfactory operation at high temperatures. The structure of the double diffused transistor is susceptible to design 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 analysis in a fashion similar to that which has been applied to other junc- tion transistors. The non-uniform distribution of impurities produces significant electrical effects which can be controlled to enhance appre- cial)ly the high-frequency behavior of the devices. The extreme control inherent in the use of diffusion to distribute im- purities in a semiconductor structure suggests that this technique will become one of the most valuable in the fabrication of semiconductor devices. ACKNOWLEDGEMENT The authors are indebted to several people who contributed to the work described in this paper. In particular, the double diffused silicon from which the transistors were prepared was supplied by C. S. Fuller and J. A. Ditzenberger. The data on diffusion coefficients and concen- trations were also obtained by them. P. W. Foy and G. Kaminsky assisted in the fabrication and mounting of the transistors and J. M. Klein aided in the electrical characterization. The computations of the various solutions of the diffusion equation, (4.3), were performed by Francis Maier. In addition many valuable discussions with C. A. Lee, G. Weinreich, J. L. Moll, and G. C. Dacey helped formu- late many of the ideas presented herein. A High-Frequency Diffused Base Gernianiuni Transistor By CHARLES A. LEE (Manuscript received November 15, 1955) Techniques of impurity diffusion and alloying have been developed which make possible the construction of p-n-p junction transistors utilizing a diffused surface layer as a base region. An important Jeature is the high degree of dimensional control obtainable. Diffusion has the advantages of being able to produce uniform large area junctions which may be utilized in high power devices, and very thin surface layers which may be utilized in high-frequency devices. Transistors have been made in germanium which typically have alphas of 0.98 and alpha-cutoff frequencies of 500 mcls. The fabrication, electrical characterization, and design considerations of these transistors are dis- cussed. INTRODUCTION Recent work ■ concerning diffusion of impurities into germanium and silicon prompted the suggestion that the dimensional control in- herent in these processes be utilized to make high-frecjuency transistors. One of the critical dimensions of junction transistors, which in many cases seriously restricts their upper freciuency limit of operation, is the thickness of the base region. A considerable advance in transistor proper- ties can be accomplished if it is possible to reduce this dimension one or two orders of magnitude. The diffusion constants of ordinary donors and acceptors in germanium are such that, with'n realizable tempera- tures and times, the depth of diffused surface layers may be as small as 10" cm. Already in the present works layers slightly less than 1 micron (10~ cm) thick have been made and utilized in transistors. Moreover, the times and temperatures required to produce 1 micron surface laj^ers permit good control of the depth of penetration and the concentration of the diffusant in the surface layer with techniciues described below. If one considers making a transistor whose base region consists of such 23 24 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 a diffused surface layer, several problems become immediately apparent : (1) Control of body resistivity and lifetime during the diffusion heat- ing cycle. (2) Control of the surface concentration of the diffusant. (3) INIaking an emitter on the surface of a thin diffused layer and controlling the depth of penetration. (4) Making an ohmic base contact to the diffused surface layer. One approach to the solution of these problems in germanium which has enabled us to make transistors with alpha-cutoff frequencies in excess of 500 mc/sec is described in the main body of the paper. An important characteristic feature of the diffusion technique is that it produces an impurity gradient in the base region of the transistor. This impurity gradiant produces a "built-in" electric field in such a direction as to aid the transport of minority carriers from emitter to collector. Such a drift field may considerably enhance the frequency response of a transistor for given physical dimensions. The capabilities of these new techniques are only partially realized by their application to the making of high frequency transistors, and even in this field their potential has not been completely explored. For example, with these techniques applied to making a p-n-i-p structure the possibility of constructing transistor amplifiers with usable gain at frequencies in excess of 1,000 mc/sec now seems feasible. DESCRIPTION OF TRANSISTOR FABRICATION AND PHYSICAL CHARACTERIS- TICS As starting material for a p-n-p structure, p-type germanium of 0.8 ohm-cm resistivity was used. From the single crystal ingot rectangular bars were cut and then lapped and polished to the approximate dimen- sions: 200 X 60 X 15 mils. After a slight etch, the bars were washed in deionized water and placed in a vacuum oven for the diffusion of an n-type impurity into the surface. The vacuum oven consisted of a small molybdenum capsule heated by radiation from a tungsten coil and sur- rounded by suitable radiation shields made also of molybdenum. The capsule could be baked out at about 1,900°C in order that impurities detrimental to the electrical characteristics of the germaniinn be evapo- rated to sufficiently low levels. As a source of n-type impurity to be placed with the p-type bars in the molybdenum oven, arsenic doped germanium was used. The rela- tively high vapor pressure of the arsenic was reduced to a desirable range (about lO"* nun of Ilg) by diluting it in germanium. The use of ger- manium eliminated any additional problems of contamination by the A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 25 dilutant, and provided a convenient means of determining the degree of dilution by a measurement of the conductivity. The arsenic concentra- tions used in the source crystal were typically of the order of 10 '-10^^/cc. These concentrations were rather high compared to the concentrations desired in the diffused surface layers since compensation had to be made for losses of arsenic due to the imperfect fit of the cover on the capsule and due to some chemical reaction and adsorption which occurred on the internal surfaces of the capsule. The layers obtained after diffusion were then evaluated for sheet con- ductivity and thickness. To measure the sheet conductivity a four-point probe method^ was used. An island of the surface layer was formed by masking and etching to reveal the junction between the surface layer and the p-type body. The island was then biased in the reverse direction with respect to the body thus effectively isolating it electrically during the measurement of its sheet conductivity. The thickness of the surface layer was obtained by first lapping at a small angle to the original surface (3^-2°~l°) and locating the junction on the beveled surface with a thermal probe; then multiplying the tangent of the angle between the two sur- faces by the distance from the edge of the bevel to the junction gives the desired thickness. Another particularly convenient method of measuring the thickness' is to place a half silvered mirror parallel to the original sur- face and count fringes, of the sodium D-Yme for example, from the edge of the bevel to the junction. Typically the transistors described here were prepared from diffused layers with a sheet conductivity of about 200 ohms/square, and a layer thickness of (1.5 ± 0.3) X 10~ cm. When the surface layer had been evaluated, the emitter and base con- tacts were made using techniques of vacuum evaporation and alloying. o For the emitter, a film of aluminum approximately 1,000 A thick was evaporated onto the surface through a mask which defined an emitter area of 1 X 2 mils. The bar with the evaporated aluminum was then placed on a strip heater in a hydrogen atmosphere and momentarily brought up to a temperature sufficient to alloy the alimiinum. The emitter having been thus formed, the bar was again placed in the masking jig and a film of gold-antimony alloy from 3,000 to 4,000 A thick was evaporated onto the surface. This film was identical in area to the emitter, and was placed parallel to and 0.5 to 1 mil away from the emitter. The bar was again placed on the heater strip and heated to the gold-germanium eutectic temperature, thus forming the ohmic base contact. The masking jig was constructed to permit the simultaneous evaporation of eight pairs of contacts on each bar. Thus, using a 3-mil diamond saw, a bar could be cut into eight units. 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Each unit, with an alloyed emitter and base contact, was then soldered to a platinum tab with indium, a sufficient quantity of indium being- used to alloy through the n-type surface layer on the back of the unit. One of the last steps was to mask the emitter and base contacts with a 6- to 8-mil diameter dot of wax and form a small area collector junction by etching the unit attached to the platinum tab, in CP4. After washing in solvents to remove the wax, the unit was mounted in a header designed to allow electrolytically pointed wire contacts to be made to the base and emitter areas of the transistor. These spring contacts were made of 1-mil phosphor bronze wire. ELECTRICAL CHARACTERIZATION Of the parameters that characterize the performance of a transistor, one of the most important is the short circuit current gain (alpha) ver- sus frequency. The measured variation of a and q:/(1 — a) (short-circuit current gain in the grounded emitter circuit) as a function of frequency for a typical unit is shown in Fig. 1 . For comparison the same parameters for an exceptionally good unit are shown in Fig. 2. In order that the alpha-cutoff frequency be a measure of the transit time of minority carriers through the active regions of the transistor, any resistance-capacity cutoffs, of the emitter and collector circuits, must lie considerably higher than the measured /„ . In the emitter circuit, an external contact resistance to the aluminum emitter of the order of 10 U1 _J LU eg o lij Q •4U ( 30 20 >-( — , 4.3 MC UNIT 0-3 p- n-p Ge Ie = 2 MA Vc =-10 VOLTS ao= 0.982 ' 1 s S. 1-a 6 DB OCT/> PER ^' VE ■> ^s 1 -10 l«l w > \ 46 3 M( 1 ; ^ * 0.1 0.2 0.4 0,6 1 2 4 6 8 10 20 40 60 100 200 400 1000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 1 — The grounded emitter and grounded base response versus frequency for a typical unit. A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 27 40 30 10 _l LU 5 20 o LJJ Q 10 o- ^ « 3.4 M C UNIT M-2 p-r Ie = 2MA 1-p Ge N -— oc 1 \-oc Vc=-10 VOLTS OCo- 0.980 6Db\ PER A OCTAVE ^N ^'s oc i-C v^ 540 MC ^\ \ -10 0.1 0.2 0.4 0.6 1 2 4 6 e 10 20 40 60 100 200 400 1000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 2 — The grounded emitter and grounded base response versus frequency for an exceptionally good unit. to 20 ohms and a junction transition capacity of 1 fx^xid were measured. The displacement current which flows through this transition capacity reduces the emitter efficiency and must be kept small relative to the injected hole current. With 1 milliampere of ciu"rent flowing through the emitter junction, and conseciuently an emitter resistance of 26 ohms, I the emitter cutoff for this transistor was above 6,000 mc/sec. One can now see that the emitter area must be small and the current density high to attain a high emitter cutoff freciuency. The fact that a low base resistance requires a high level of doping in the base region, and thus a high emitter transition capacity, restricts one to small areas and high current densities. In the collector circuit capacities of 0.5 to 0.8 ^l^xid at a collector volt- age of — 10 volts were measured. There was a spreading resistance in the collector body of about 100 ohms which was the result of the small emitter area. The base resistance was approximately 100 ohms. If the phase shift and attenuation due to the transport of minority carriers through the base region w^ere small at the collector cutoff frequency, the (effective base resistance would be decreased by the factor (1 —a). The collector cutoff frequency is then given by where Cc = collector transition capacity and Re = collector body spreading resistance. 28 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 However, in the transistors described here the base region produces the major contribution to the observed alpha-cutoff frequency and it is more appropriate to use the expression 2irCcin + Re) where n = base resistance. This cutoff frequency could be raised by in- creasing the collector voltage, but the allowable power dissipation in the mounting determines an upper limit for this voltage. It should b noted that an increase in the doping of the collector material would raise the cutoff since the spreading resistance is inversely proportional to Na , while the junction capacity for constant collector voltage is only pro- portional to Na . The low-frequency alpha of the transistor ranged from 0.95 to 0.99 with some exceptional units as high as 0.998. The factors to be con- sidered here are the emitter efficiency y and the transport factor (3. The transport factor is dependent upon the lifetime in the base region, the recombination velocity at the surface immediately surrounding the emitter, and the geometry. The geometrical factor of the ratio of the emitter dimensions to the base layer thickness is > 10, indicating that solutions for a planar geometry may be assumed.^ If a lifetime in the base region of 1 microsecond and a surface recombination velocity of 2,000 cm/sec is assumed a perturbation calculation gives iS = 0.995 The high value of ^ obtained with what is estimated to be a low base region lifetime and a high surface recombination velocity indicates that the observed low frecjuency alpha is most probably limited by the emitter injection efficiency. As for the emitter injection efficiency, within the accuracy to which the impurity concentrations in the emitter re- growth layer and the base region are known, together with the thick- nesses of these two regions, the calculated efficiency is consistent with the experimentally observed values. Considerations of Transit Time An examination of what agreement (^xists between the alpha-cutoff frequency and the physical measurements of the base region involves the me(;hanism of transport of minority carriers through the active regions of the transistor. The "active regions" include the space charge A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 29 region of the collector junction. The transit time through this region is no longer a negligible factor. A short calculation will show that with — 10 volts on the collector junction, the space charger layer is about 4 X 10"^ cm thick and that the frequency cutoff associated with trans- port through this region is approximately 3,000 mc/sec. The remaining problem is the transport of minority carriers through the base region. Depending upon the boundary conditions existing at the surface of the germanium during the diffusion process, considerable gradients of the impurity density in the surface layer are possible. How- ever, the problem of what boundary conditions existed during the diffu- sion process employed in the fabrication of these transistors w^ill not be discussed here because of the many uncertainties involved. Some quali- tative idea is necessary though of how electric fields arising from impurity gradients may affect the frequency behavior of a transistor in the limit of low injection. If one assumes a constant electric field as would result from an ex- ponential impurity gradient in the base region of a transistor, then the continuity eciuation may be solved for the distribution of minority carriers.* From the hole distribution one can obtain an expression for the transport factor j3 and it has the form /3 = e" r? sinh Z -{- Z cosh Z where 1, Ne IqE ^"2^^iV; = 2^^' z ^ [i^ + ,r' IV' Ne = donor density in base region at emitter junction Nc = donor density in base region at collector junction E = electric field strength Dp = diffusion constant for holes w = width of the base layer 30 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 A plot of this function for various values of rj is shown in Fig. 3. For ?? = 0, the above expression reduces to the well known case of a uniformly doped base region. The important feature to be noted in Fig. 3 is that relatively small gradients of the impurity distribution in the base layer can produce a considerable enhancement of the frequency response. It is instructive to calculate what the alpha-cutoff f recjuency would be for a base region with a uniform distribution of impurity. The effective thickness of the base layer may be estimated by decreasing the measured thickness of the surface layer by the penetration of the space charge region of the collector and the depth of the alloyed emitter structure. Using a value for the diffusion constant of holes in the base region appro- priate to a donor density of about 10 Vcc, 300 mc/s ^fa^ 800 mc/s This result implies that the frecjuency enhancement due to "built-in" fields is at most a factor of two. In addition it was observed that the alpha-cutoff frequency was a function of the emitter current as shown in Fig. 4. This variation indicates that at least intermediate injection <Si £L '^ ^ 77 siNhZ +Z coshz Z=(L5z5+772)'/2 0.8 0.6 0.4 ' > *~ :;^;~->i^ k.^ ^ Nv ^ N "\ V \ \ \ V ^ \ \ >v, 0.2 A \ \ K \ \ \ i \ K \ \ \ 0.08 - ^ \— ^ A — \ — v\- 0.06 0.04 - ^^ \ ^, ^ \ ^ 1\ 4i r 0.02 \ \ \ V \ \ V 0.01 1 1 1 \ \ 1 1 > 1 1 1 1 _L 0.1 0.2 0.4 0.6 0.8 1 6 8 10 20 40 60 80 100 w2 <^-U} -g- , (RADIANS) Fig. .3 — The variation of | i3 | ver.sii.s frequency for various values of a uniform drift field in the base region. A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 31 in _i LU m o LU a z b n =7^" '^^-^^ S— 1' i f \ ^ ' ' ; Q ■ ■_;;;; -t Fv Rl k -5 UNIT 0-3 p-n-p Ge o Ie = 2 MA A Ie=0.8MA D Ig=0.4MA \ k^ ^ \ \ \ 10 Vc = -K ) VOLTS 1 1 \ 1 1 1 10 20 30 40 50 60 80 100 200 300 400 FREQUENCY IN MEGACYCLES PER SECOND 600 800 1000 Fig. 4 current. The variation of the alpha-cutoff frequency as a function of emitter levels exist in the range of emitter current shown in Fig. 4. The conclu- sion to be drawn then is that electric fields produced by impurity gradients in the base region are not the dominant factor in the transport of minority carriers in these transistors. The emitter current for a low level of injection could not be deter- mined by measuring /„ versus /« because the high input impedance at very low levels was shorted by the input capacity of the header and socket. Thus at very small emitter currents the measured cutoff fre- quency was due to an emitter cutoff and was roughly proportional to the emitter current. At /e ^ 1 ma this effect is small, but here at least intermediate levels of injection already exist. A further attempt to measure the effect of any "built-in" fields by turning the transistor around and measuring the inverse alpha proved fruitless for two reasons. The unfavorable geometrical factor of a large collector area an a small emitter area as well as a poor injection effi- ciency gave an alpha of only a = 0.1 Secondly, the injection efficiency turns out in this case to be proportional to oT^^'^ giving a cutoff freciuency of less than 1 mc/sec. The sciuare-root dependence of the injection efficiency on freciuency may be readily seen. The electron current injected into the collector body may be expressed as Je = qDnN 1 -)- iu^Te 1/2 where q = electronic charge 32 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Dn ^ diffusion constant of electrons Vi = voltage across collector junction Tic = density of electrons on the p-type side of the collector junction Te = lifetime of electrons in collector body Le = diffusion length of electrons in the collector body Since the inverse cutoff frequency is well below that associated with the base region, we may regard the injected hole current as independent of the frequency in this region. The injection efficiency is low so that 7 ;^ ^ « 1 J e Thus at a frequency where then cor, »1 I -1/2 An interesting feature of these transistors was the very high current densities at which the emitter could be operated without appreciable loss of injection efficiency. Fig. 5 shows the transmission of a 50 millimicro- second pulse up to currents of 18 milliamperes which corresponds to a current density of 1800 amperes/cm". The injection efficiency should remain high as long as the electron density at the emitter edge of the base region remains small compared to the acceptor density in the emitter regrowth layer. When high injection levels are reached the in- jected hole density at the emitter greatly exceeds the donor density in th(> base region. In order to preserve charge neutrality then p ^ n where p = hole density n = electron density As the inject(Hl hole density is raised still further the electron density will eventually become comparable to the acceptor density in the emitter regrowth layer. Tlie density of acceptors in the emitter regrowth A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 33 30 46 60 75 90 TIME IN MILLIMICROSECONDS >" 9 "^ V 4 '^ \^ / 18 V / -15 15 30 45 60 75 90 TIME IN MILLIMICROSECONDS 105 120 136 Fig. 5 — Transmission of a 50 millimicrosecond pulse at emitter currents up to 18 ma by a typical unit. (Courtesy of F. K. Bowers). region is of the order of and this is to be compared with injected hole density at the base region iside of the emitter junction. The relation between the injected hole density and the current density may be approximated by^ J. w where pi = hole density at emitter side of base region w = width of base region jA short calculation indicates that the emitter efficiency should remain 'high at a current density of an order of magnitude higher than 1,800 |amp/cm'. The measurements were not carried to higher current densities jbecause the voltage drop across the spreading resistance in the collector was producing saturation of the collector junction. CONCLUSIONS Impurity diffusion is an extremely powerful tool for the fabrication of high frequency transistors. Moreover, of the 50-odd transistors which 34 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 were made in the laboratory, the characteristics were remarkably uni- form considering the ^•ariations usually encountered at such a stage of development. It appears that diffusion process is sufficiently controllable that the thickness of the base region can be reduced to half that of the units described here. Therefore, with no change in the other design parameters, outside of perhaps a different mounting, units with a 1000 mc/s cutoff frequency should be possible. ACKNOWLEDGMENT The author wishes to acknowledge the help of P. W. Foy and W. Wieg- mann who aided in the construction of the transistors, D. E. Thomas who designed the electrical equipment needed to characterize these units, and J. Klein who helped with the electrical measurements. The numerical evaluation of alpha for drift fields was done by Lillian Lee whose as- sistance is gratefully acknowledged. REFERENCES 1. C. S. Fuller, Phys. Rev., 86, pp. 136-137, 1952. 2. J. Saby and W. C. Dunlap, Jr., Phys. Rev., 90, p. 630, 1953. 3. W. Shocklej', private communication. 4. H. Kromer, Archiv. der Elek. tlbertragung, 8, No. 5, pp. 223-228, 1954. 5. R. A. Logan and M. Schwartz, Phys. Rev., 96, p. 46, 1954 6. L. B. Valdes, Proc. I.R.E., 42, pp. 420-427, 1954. 7. W. L. Bond and F. M. Smits, to be published. 8. E. S. Rittner, Pnys. Rev., 94, p. 1161, 1954. 9. W. M. Webster, Proc. I.R.E., 42, p. 914, 1954. 10. J. M. Early, B.S.T.J., 33, pp. 517-533, 1954. Waveguide Investigations with Millimicrosecond Pulses By A. C. BECK (Manuscript received October 11, 1955) Pulse techniques have been used for many waveguide testing 'puryoses. The importance of increased resolution hy means of short pulses has led to the development of equipment to generate, receive and display pidses about 5 or 6 millimicroseconds lo7ig. The equipment is briefly described and its resolution and measuring range are discussed. Domi7ia7it mode waveguide and antenna tests are described, and illustrated. Applications to midtimode waveguides are then considered. Mode separation, delay distortion and its equalization, and mode conversion are discussed, and examples are given. The resolution obtained with this equipment provides information that is difficult to get by any other means, and its use has proved to be very helpfid in ivaveguide investigations. CONTENTS 1 . Introduction 35 2. Pulse Generation 36 3. Receiver and Indicator 41 4. Resolution and Measuring Range 42 5. Dominant Mode Waveguide Tests 43 6. Testing Antenna Installations 45 7. Separation of Modes on a Time Basis 48 8. Delay Distortion 52 9. Delay Distortion Ecjualization 54 10. Measuring Mode Conversion from Isolated Sources 57 11. Measuring Distril)uted Mode Conversion in 1 ong Waveguides 61 12. Concluding Remarks 65 1. INTRODUCTION Pulse testing techniques have been employed to advantage in wave- guide investigations in numerous ways. The importance of better resolu- tion through the use of short pulses has always been apparent and, from the first, eciuipment was employed which used as short a pulse as pos- sible. Radar-type apparatus using magnetrons and a pulse width of about one-tenth microsecond has seen considerable use in waveguide research, and many of the results have been published.' • - 35 36 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 To improve the resolution, work was initiated some time ago by S. E. Miller to obtain measuring equipment which would operate with much shorter pulses. As a result, pulses about 5 or 6 millimicroseconds long became available at a frequency of 9,000 mc. In a pulse of this length there are less than 100 cycles of radio frequency energy, and the signal occupies less than ten feet of path length in the transmission medium. The RF bandwidth required is about 500 mc. In order to obtain such bandwidths, traveling wave tubes were developed by J. R. Pierce and members of the Electronics Research Department of the Laboratories. The completed amplifiers were designed by W. W. Mumford. N. J. Pierce, R. W. Dawson and J. W. Bell assisted in the design and construc- tion phases, and G. D. Mandeville has been closely associated in all of this work. 2. PULSE GENERATION These millimicrosecond pulses have been produced by two different types of generators. In the first equipment, a regenerative pulse gener- ator of the type suggested by C. C. Cutler of the Laboratories was used.^ This was a very useful device, although somewhat complicated and hard to keep in adjustment. A brief description of it will permit comparisons with a simpler generator which was developed a little later. A block diagram of the regenerative pulse generator is shown in Fig. 1. The fundamental part of the system is the feedback loop drawn with heavy lines in the lower central part of the figure. This includes a travel- ing wave amplifier, a waveguide delay line about sixty feet long, a crystal expander, a band-pass filter, and an attenuator. This combination forms an oscillator which produces very short pulses of microwave energy. Between pulses, the expander makes the feedback loop loss too high for oscillation. Each time the pulse circulates around the loop it tends to shorten, due to the greater amplification of its narrower upper part caused by the expander action, until it uses the entire available band width. A 500-mc gaussian band-pass filter is used in the feedback loop,^ of this generator to determine the final bandwidth. An automatic gain control operates with the expander to limit the pulse amplitude, thus preventing amplifier compression from reducing the available expansion. To get enough separation between outgoing pulses for reflected pulse measurements with waveguides, the repetition rate would need to be too low for a practical delay fine length in the loop. Therefore a r2.8-mc fundamental rate was chosen, and a gated traveling wave {\\\)v ampfifier was used to reduce it to a 100-kc rate at the output. This amplifier is kept in a cutoff condition for 127 pulses, and then a gate pulse restores I i t WAVEGUIDE TESTING WITH MILLIMIf'ROSECOND PULSES 37 it to the normal amplifying condition for fifty millimicroseconds, during which time the 128th pulse is passed on to the output of the generator as shown on Fig. 1. The synchronizing system is also shown on Fig. 1. A 100-kc quartz crystal controlled oscillator with three cathode follower outputs is the basis of the system. One output goes through a seven stage multiplier to get a 12.8-mc signal, which is used to control a pulser for synchroniz- ing the circulating loop. Another output controls the gate pulser for the output traveling wave amplifier. Accurate timing of the gate pulse is obtained by adding the 12.8-mc pulses through a buffer amplifier to the gate pulser. The third output synchronizes the indicator oscilloscope sweep to give a steady pattern on the screen. Although this equipment was fairly satisfactory and served for many OSCILLATOR AND CATHODE FOLLOWERS 100 KC I 1 MULTIPLIER 100 KC TO 12.8 MC SYNC PULSER 0.02 A SEC 12.8 MC 500 MC BANDPASS FILTER GATE PULSER 0.05 USEC 100 KC A BUFFER AMPLIFIER "1 CRYSTAL EXPANDER U AGC I WAVEGUIDE DELAY LINE TW TUBE ■Y^ MILLI/iSEC/ 9000 MC/' PULSES 12.8 MC RATE MlLLIyUSEC 9000 MC PULSES 100 KC RATE GATED TW TUBE SYNC SIGNAL TO INDICATOR SCOPE Fig. 1 — Block diagram of the regenerative pulse generator. 38 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 testing purposes, it was rather complex and there were some problems in its construction and use. It was difficult to obtain suitable microwave crystals to match the waveguide at low levels in the expander. Tliis would make it even more difficult to build this type of pulse generator for higher frequency ranges. Stability also proved to be a problem. The frequency multiplier had to be very well constructed to avoid phase shift due to drifting. The gate pulser also required care in design and construction in order to get a stable and flat output pulse. It was some- what troublesome to keep the gain adjusted for proper operation, and the gate pulse time adjustment required some attention. The pulse frequency could not be changed. For these reasons, and in order to get a smaller, lighter and less complicated pulse generator, work was carried out to produce pulses of about the same length by a simpler method. If the gated output amplifier of Fig. 1 were to have a CW instead of a pulsed input, a pulse of microwave energy would nevertheless appear at the output because of the presence of the gating pulse. This gating pulse is applied to the beam forming electrode of the tube to obtain the gating action. If the beam forming electrode could be pulsed from cutoff to its normal operating potential for a very short time, very short pulses of output energy could be obtained from a continuous input signal. How- ever, it is difficult to obtain millimicrosecond video gating pulses of suf- ficient amplitude for this purpose at a 100-kc repetition rate. A traveling-wave tube amplifies normally only when the helix is within a small voltage range around its rated dc operating value. For voltages either above or below this range, the tube is cut off. When the helix voltage is raised through this range into the cutoff region beyond it, and then brought back again, two pulses are obtained, one during a small part of the rise time and the other during a small part of the return time. If the rise and fall times are steep, very short pulses can be obtained. Fig. 2 shows the pulse envelopes photographed from the indicator scope screen when this is done. For the top trace, the helix was biased 300 volts negatively from its normal operating potential, then pulsed to its correct operating range for about 80 millimicroseconds, during which time normal amplification of the CW input signal was ob- tained. The effect of further increasing the helix video pulse amplitude in the positive direction is shown by the succeeding lower traces. The envelope dips in the middle, then two separated pulses remain — one during a part of the rise time and one during a part of the fall time of helix voltage. The pulses shown on the bottom trace have shortened to about six millimicroseconds in length. The helix pulse had a positive amplitude of about 500 volts for this trace. 1 WAVEGUIDE TESTIXG WITH MILUMICROSErOXD PULSES 39 Since only one of these pulses can be used to get the desired repetition rate, it is necessary to eliminate the other pulse. This is done in a simi- lar manner to that used for gating out the undesired pulses in the re- generative pulse generator. However, it is not necessary to use another amplifier, as was required there, since the same tube can be used for this purpose, as well as for producing the microwave pulses. Its beam forming electrode is biased negatively about 250 volts with respect to the cathode, and then is pulsed to the normal operating potential for about 50 millimicroseconds during the time of the first short pulse ob- tained by gating the helix. Thus, the beam forming electrode potential has been returned to the cutoff value during the second helix pulse, which is therefore eliminated. Il A block diagram of the resulting double-gated pulse generator is shown in Fig. 3. Comparison with Fig. 1 shows that it is simpler than the regenerative pulse generator, and it has also proved more satisfactory in operation. It can be used at any frequency where a sig- nal source and a traveling-wave amplifier are available, and the pulse Fig. 2 — Envelopes of microwave pulses at the output of a traveling wave am- lifier with continuous wave input and helix gating. The gating voltage is higher or the lower traces. 40 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 frequency can be set anywhere within the bandwidth of the travehng- wave ampUfier by tuning the klystron oscillator. The pulse center frequency is shifted from that of the klystron os- cillator frequency by this helix gating process. An over-simphfied but helpful explanation of this effect can be obtained by considering that the microwave signal voltage on the helix causes a bunching of the elec- tron stream. This^ bunching has the same periodicity as the microwave signal voltage when the dc helix potential is held constant. However, since the helix voltage is continuously increased in the positive direction during the time of the first pulse, the average velocity of the last bunches of electrons becomes higher than that of the earlier bunches in the pulse, because the later electrons come along at the time of higher positive helix voltage. This tends to shorten the total length of the series of bunches, resulting in a shorter w^avelength at the output end of the helix and therefore a higher output microwave frequency. On the second pulse, obtained when the helix voltage returns toward zero, the process is reversed, the bunching is stretched out, and the frequency is de- creased. This second pulse is, however, gated out in this arrangement by the beam-forming electrode pulsing voltage. The result for this particular tube and pulse length is an effective output frequency ap- proximately 150 mc higher than the oscillator frequency, but this figure is not constant over the range of pulse frequencies available within the amplifier bandwidth. OSCILLATOR AND CATHODE FOLLOWERS 100 KC KLYSTRON OSCILLATOR 9000 MC BEAM FORMING ELECTRODE PULSER HELIX PULSER ^ PULSED TW TUBE MILLI/aSEC 9000 MC PULSES SYNC SIGNAL TO INDICATOR SCOPE Fig. 3 — Block diugram of the double-gated traveling wave tube millimicro- second pulse generator. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 41 3. RECEIVER AND INDICATOR The receiving equipment is shown in Fig. 4. It uses two traveUng- wave amplifiers in cascade. A wide band detector and a video amplifier then follow, and the signal envelope is displayed by connecting it to the vertical deflecting plates of a 5 XP type oscilloscope tube. The video amplifier now consists of two Hewlett Packard wide band dis- tributed amplifiers, having a baseband width of about 175 mc. The second one of these has been modified to give a higher output voltage. The sweep circuits for this oscilloscope have been built especially for this use, and produce a sweep speed in the order of 6 feet per micro- second. An intensity pulser is used to eliminate the return trace. These parts of the system are controlled by a synchronizing output from the pulse generator 100-kc oscillator. A precision phase shifter is used at the receiver for the same purpose that a range unit is employed in radar systems. This has a dial, calibrated in millimicroseconds, which moves the position of a pulse appearing on the scope and makes accurate measurement of pulse delay time possible. Fig. 4 also shows the appearance of the pulses obtained with this equipment. The pulse on the left-hand side of this trace came from the PULSE SIGNAL 9000 MC SYNC SIGNAL 100 KC TW TUBES VIDEO AMPLIFIER INTENSITY PULSER 0.05/USEC 100 KC PRECISION PHASE SHIFTER SWEEP GENERATOR DOUBLE-GATED PULSE REGENERATIVE PULSE Fig. 4 — Block diagram of millimicrosecond pulse receiver and indicator. The idicator trace photograph shows pulses from each type of generator. 42 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 newer double-gated pulse generator, while the pulse on the right was produced by the regenerative pulse generator. It can be seen that they appear to have about the same pulse width and shape. This is partly due to the fact that the video amplifier bandwidth is not c^uite adequate to show the actual shape, since in both cases the pulses are slightly shorter than can be correctly reproduced through this amplifier. The ripples on the base line following the pulses are also due to the video amplifier characteristics when used with such short pulses. 4. RESOLUTION AND MEASURING RANGE Fig. 5 shows a piece of equipment which was placed between the pulse generator and the receiver to show the resolution which can be obtained. This waveguide hybrid junction has its branch marked 1 connected to the pulse generator and branch 3 connected to the receiver. If the two side branches marked 2 and 4 were terminated, substantially no energy would be transmitted from the pulser straight through to the receiver. However, a short circuit placed on either side branch will send energy through the system to the receiver. Two short circuits were so placed that the one on branch 4 was 4 feet farther away from the hybiid junc- tion than the one on branch 2. The pulse appearing first is produced l)y a signal traveling from the pulse generator to the short circuit on branch 2 and then through to the receiver, as shown by the path drawn with short dashes. A second pulse is produced by the signal which travels BRANCH 2 SHORT CIRCUIT BRANCH FROM PULSER TO RECEIVER FIRST PULSE PATH SECOND PULSE PATH SHORT CIRCUIT DOUBLE-GATED PULSES REGENERATIVE PULSES Fig. 5 — W;iv(!guicle hyhriil ciicuil- uscxl to demonstrate resululion of milli- microsecond pulses. Trace photographs of pulses from each type of generator ;iie shown. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 43 TO RECEIVER \ TE° IN 3"DIAM copper GUIDE (ISO FT LONG) Fig. 6 — Waveguide arrangement and oscilloscope trace photos showing pres- ence and location of defective joint. The dominant mode (TEn) was used with its polarization changed 90 degrees for the two trace photos. from the pulse generator through branch 4 to the short circuit and then to the receiver as shown by the long dashed line. This pulse has traveled 8 feet farther in the waveguide than the first pulse. This would be equiva- lent to seeing separate radar echoes from two targets about 4 feet apart. Resolution tests made in this way \vith the pulses from the regenerative pulse generator, and from the double-gated pulse generator, are shown on Fig. 5. With our video amplifier and viewing equipment, there is no appreciable difference in the resolution obtained using either type of pulse generator. The measuring range is determined by the power output of the gated amplifier at saturation and by the noise figure of the first tube in the receiver. In this equipment the saturation level is about 1 watt, and the noise figure of the first receiver tube is rather poor. As a result, received pulses about 70 db below the outgoing pulse can be observed, which is I enough range for many measurement purposes. 5. DOMINANT MODE WAVEGUIDE TESTS Fig. 6 shows the use of this equipment to test 3'^ round waveguides such as those installed between radio repeater equipment and an an- tenna. This particular 150-foot line had very good soldered joints and was thought to be electrically very smooth. The signal is sent in through a transducer to produce the dominant TEn mode. The receiver is con- nected through a directional coupler on the sending end to look for any 44 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 7 — Defective joint caused by imperfect soldering which gave the reflec- tion shown on Fig. 6. reflections from imperfections in the line. The overloaded signal at the left of the oscilloscope trace is produced by leakage directly through the directional coupler. The overloaded signal on the other end of this trace is produced by the reflection from the short circuit piston at the far end of the waveguide. The signal between these two, which is about 45 db down from the input signal, is produced by an imperfect joint in the waveguide. The signal polarization was oriented so that a maxi- mum reflection was obtained in the case of the lower trace. In the other trace, the polarization was changed by 90°. It is seen that this particular joint produces a stronger reflection for one polarization than for the other. By use of the precision phase shifter in the receiver the exact location of this defect was found and the particular joint that was at fault was sawed out. Fig. 7 shows this joint after the pipe had been cut in half through the middle. The guide is quite smooth on the inside in spite of the discoloration of some solder that is shown here, but on the left-hand side of the illustration the open crack is seen where the solder did not run in properly. This causes the reflected pulse that shows on the trace. The fact that this crack is less than a semi-circumference in length causes the echo to be stronger for one polarization than for the other. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 45 Fig. 8 shows the same test for a 3" diameter ahiminum waveguide 250 feet long. This line was mounted horizontally in the test building with compression couplings used at the joints. The line expanded on warm days hut the friction of the mounting supports was so great that it pulled open at some of the joints when the temperature returned to normal. These open joints produced reflected pulses from 40 to 50 db down, which are shown here. They come at intervals equal to the length of one section of pipe, about 12 feet. Some of these show polarization effects where the crack was more open on one side than on the other, but others are almost independent of polarization. These two photo- graphs of the trace were taken with the polarization changed 90°. Fig. 9 shows the same test for a 3" diameter galvanized iron wave- guide. This line had shown fairly high loss using CW for measure- ments. The existence of a great many echoes from random distances indicates a rough interior finish in the waveguide. Fig. 10 shows the kind of inperfections in the zinc coating used for galvanizing which caused these reflections. 6. TESTING ANTENNA INSTALLATIONS The use of this equipment in testing waveguide and antenna installa- tions for microwave radio repeater systems is shown in Fig. 11. This particular work was done in cooperation wdth A. B. Crawford's antenna research group at Holmdel, who designed the antenna system. A direc- tional coupler was used to observe energy reflections from the system under test. In this installation a 3" diameter round guide carrying the TEu mode was used to feed the antenna. Two different waveguide TE,, IN 3"D1AM aluminum GUIDE (250 FT LONG) Fig. 8 — Reflections from several defective joints in a dominant (TEn) mode waveguide. The two trace photos are for polarizations differing by 90 degrees. 46 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 TO RECEIVER ^n ^— a^IS^rv^ i— ^ TE ■^-^Bi 21 I 10 TE,° IN 3" DIAM GALVANIZED IRON GUIDE (250 FT LONG) Fig. 9 ■ — Multiple reflections from a dominant (TEn) mode waveguide with a rough inside surface. The two trace photos are for polarizations differing by 90 I degrees. joints are shown here. In addition, a study was being made of the re- turn loss of the transition piece at the throat of the antenna which • connected the 3" waveguide to the square section of the horn. The I waveguide sections are about 10 feet long. The overloaded pulse at the left on the traces is the leakage through the directional coupler. The Fig. 10 — Rough inside surface of a galvanized iron waveguide which produced the reflections shown on Fig. 9. I WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 47 other echoes are associated with the parts of the system from which they came by the dashed Hues and arrows on the figure. A clamped joint in the line gave the reflection shown next following the initial overloaded pulse. A well made threaded coupling in which the ends of the pipe butted squarel,y is seen to have a very much lower reflection, scarcely observable on this trace. Since there is ahvays reflection from the mouth and upper reflector parts of this kind of antenna, it is not possible to measure a throat transition piece alone by conventional CW methods, as the total reflected power from the system is measured. Here, use of the resolution of this short pulse equipment completely separated the reflection of the transition piece from all other reflections and made a measurement of its performance possible. In this particular case, the reflection from the transition is more than 50 db down from the incident signal which represents very good design. As can be seen, OPEN APERTURE FIBERGLASS COVER OVER APERTURE REFLECTION APPEARS -^TO COME FROM 16 FT N FRONT OF HORN MOUTH DIRECTIONAL TRANSDUCER CLAMPED THREADED ROUND-TO COUPLER JOINT COUPLING SQUARE TRANSITION Fig. 11 — Waveguide and antenna arrangement with trace photos showing re- flections from joints, transition section, and cover. 48 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 the reflection from the parabohc reflector and mouth is also finite low, and this characterizes a good antenna installation. The extra reflected pulse on the right of the lower trace on Fig. 11 appeared when a fiberglas weatherproof cover was installed over the open mouth of the horn. This cover by itself would normally produce a troublesome reflection. However, in this antenna, it is a continuation of one of the side walls of the horn. Consequently, outgoing signals strike it at an oblique angle. Reflected energy from it is not focused by the parabolic section back at the waveguide, so the overall reflected power in the waveguide was found to be rather low. However, measuring it with this equipment, we found that an extra reflection appeared to come from a point 16 feet out in front of the mouth of the horn when the cover was in place. This is accounted for by the fact that energy re- flected obliquely from this cover bounces back and forth inside the horn before getting back into the waveguide, thus traveling the extra distance that makes the measurement seem to show that it comes from 16 feet out in front. 7. SEPARATION OF MODES ON A TIME BASIS If a pulse of energy is introduced into a moderate length of round waveguide to excite a number of modes which travel with different group velocities, and then observed farther along the line, or reflected from a piston at the end and observed at the beginning, separate pulses will be seen corresponding to each mode that is sent. This is illustrated ! t r t t TE„ TMo,TE2, TM„ TE3, (TEoi) ^NOT EXCITED TO RECEIVER =^ ^-^ =^^ t ;ft t TMj, TE4I TE,2 TM02 TM3, AND TE5, TOO WEAK TO SHOW TE, •^^ " PROBE 3 DIAM ROUND GUIDE COUPLING (WILL SUPPORT 12 MODES) Fig. 12 — Arrangement for showing mode separation on a time basis in a multi- mode waveguide. The pulses in the trace ])]io(o have all traveled to the iiisloii and back. The earlier outgoing pulse due to direelional coupler unbalance is not shown. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 49 in Fig. 12. In this arrangement energy was sent into the round line from a probe inserted in the side of the guide. This couples to all of the 12 modes which can be supported, with the exception of the TEoi circular electric mode. The sending end of the round guide was terminated. A directional coupler is connected to the sending probe so that the return from the piston at the far end can be observed on the receiver. Because of the different time that each mode takes to travel one round trip in this waveguide, which was 258 feet long, separate pulses are seen for each mode. The pulses in this figure have been marked to show which mode is being received. The time of each pulse referred to the outgoing pulse was measured and found to check very well with the calculated time. The formula for the time of transit in the waveguide for any mode is: T = 0.98322V'1 - VnJ [where T = time in millimicroseconds L = length of pulse travel in feet Vnm ^^ A /Ac X = operating wavelength in air Ac = cutoff wavelength of guide for the mode involved. [ Table I — Calculated and Measured Value of Time for One Round Trip Time in Millimicroseconds Mode Designation Calculated Measured 1 TEn 545 545 2 TMoi 561 561 3 TE,i 587 587 4 TMn 634 634 5 TEoi 634 . 6 TE31 665 665 7 TM21 795 793 8 TE4: 835 838 9 TE12 838 10 TM„2 890 890 11 TMn 1461 — 12 TE51 1519 — 50 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The calculated and measured \'alue of time for one round trip is given in Table I. In this experiment the operating wavelength was 3.35 centimeters This was obtained by measurements based on group velocit}' in a num- ber of guides as well as information about the pulse generator com- ponents. It represents an effective wa\'elength giving correct time of travel. The pulse occupies such a wide bandwidth that a measurement of its wavelength is difficult by the usual means. The dashes in the measured column indicate that the mode was not excited by the probe or was too weak to measure. These modes do not appear on the oscilloscope trace photograph. The relative pulse heights can be calculated from a knowledge of the probe coupling factors and the line loss. The probe coupling factors as given by M. Aronoff in unpublished work are expressed by the following For TE„„, modes: P = 2.390 r—^ i For TM„^ modes: TV- L a -. j\. nm ^ "flu X X P = 1.195€„ — - where P = ratio of probe coupling power in mode nm to that in mode TEn n = first index of mode being calculated Knm = Bessel function zero value for mode being calculated = Td/\c X = wavelength in air X(, = wavelength in the guide for the mode involved ' Xc = cutoff wavelength of guide for the mode involved €„ = 1 for w = €„ = 2 for n ?^ , d = waveguide diameter Formulas for guide loss as given by S. A. Schelkunoff on page 390 of his book Elect romagnelir Waves for this case where the resistivity of the aluminum guide is 4.14 X 10~^ ohms per cm cube are: WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 51 For TE„,„ modes: a = 3.805 ! — 2 2 + V.an ) (1 " Vnm) \l\n,n — n / For TM„,„ modes: a = 3.805(1 - VnJy''' where: a = attemiation of this aluminum guide in db n — first index of mode being calculated Knm — Bessel function zero value for mode being calculated = TrtZ/Xc Vnm = A/Ac X = operating wavelength in air Xc = cutoff wavelength of guide for the mode involved d = waveguide diameter Table II gives the calculated probe coupling factor, line loss, and rela- tive pulse height for each mode. In the calculation of the latter, wave elUpticity and loss due to mode conversion were neglected, but the heat loss given by the preceding formulas has been increased 20 per cent for all modes, to take account of surface roughness. Relative pulse heights were obtained by subtracting the relative line loss from twice the rela- tive probe coupling factor. The relative line loss is the number in the itable minus 2.33 db, the loss for the TEn mode. The actual pulse heights on the photo of the trace on Fig. 12 are in fair agreement with these calculated values. Differences are probably due to polarization rotation in the guide (wave ellipticity) and conver- sion to other modes, effects which were neglected in the calculations, and which are different for different modes. Calculated pulse heights with this guide length, except for modes near cutoff, vary less than the probe coupling factors, because line loss is high when tight probe coupling exists. This is to be expected, since both are the result of high fields near the guide walls. The table of round trip travel time shows that the TE41 and TE12 modes are separated by only three millimicroseconds after the round trip in this waveguide. They would not be resolved as separate pulses by this e(iuipment. However, the table of calculated pulse heights shows that the TE41 pulse should be about 22 db higher than the TE12 pulse. 52 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table II — Calculated Probe Coupling Factor, Line Loss and Pulse Height for Each Mode Mode Mode Relative Probe 1.2 X Theoretical Calculated Relatix e Number Designation Coupling Factor, db Line Loss, db Pulse Heights, db 1 TEu 2.33 2 TMoi +0.32 4.88 -1.91 3 TE2, +2.86 4.85 +3.20 4 TMu +2.80 5.51 +2.42 5 TEo, — 00 1.73 — 00 6 TE31 +4.82 8.21 +3.76 7 TM2, + 1.82 6.92 -0.95 8 TE41 +6.80 13.86 +2.07 9 TE12 -8.73 4.70 -19.83 10 TM02 -1.68 7.74 -8.77 11 TMsi -0.82 12.71 -12.02 12 TE51 + 10.14 32.09 -9.48 Since the TE12 pulse is so weak, it would not show on the trace even if it were resolved on a time basis. Coupling to the TM02 mode is rather weak, and the gain was increased somewhat at its position on the trace to show its time location. 8. DELAY distortion Another effect of the wide bandwidth of the pulses used with this equipment can be observed in Fig. 12. The pulses that have traveled for a longer time in the guide are in the modes closer to cutoff, and are on the right-hand side of the oscilloscope trace. They are broadened and distorted compared with the ones on the left-hand side. This effect is due to delay distortion in the guide. This can be explained by refer- ence to Fig. 13. On this figure the ratio of group velocity to the velocity in an unbounded medium is shown plotted as a function of frequency for each of the modes that can be propagated. The bandwidth of the transmitted pulse is indicated by the vertical shaded area. It will he noticed that the spacing of the pulses on the oscilloscope trace on Fig. 12 from left to right in time corresponds to the spacing of the group velocity curves in the bandwidth of the pulse from top to bottom. De- lay distortion on these curves is shown by the slope of the line across the pulse bandwidth. If the line were horizontal, showing the same group velocity at all points in the band, there would be no delay distortion. The greater the difference in group A-elocity at the two edges of the band, the greater the delay distortion. The curves of Fig. 13 indicate WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 53 I that there should be increasing amounts of delay distortion reading ifrom top to bottom for the pulse bandwidth used in these experiments. ;The effect of this delay distortion is to cause a broadening of the pulse. Examination of the pulse pattern of Fig. 12 shows that the later pulses corresponding in mode designation to the lower curves of Fig. 13 do in- deed show a broadening due to the increased delay distortion. One method of reducing the effect of delay distortion is to use frequency division multiplex so that each signal uses a smaller bandwidth. Another way, suggested by D. H. Ring, is to invert the band in a section of the waveguide between one pair of repeaters compared with that between an adjacent pair of repeaters so that the slope is, in effect, placed in the opposite direction, and delay distortion tends to cancel out, to a first order at least. The (luantitative magnitude of delay distortion has been expressed by S. Darlington in terms of the modulating base-band frequency needed to generate two side frequencies which suffer a relative phase error of 180° in traversing the line. This would cause cancellation of a single frequency AM signal, and severe distortion using any of the 1.0 PULSE BANDWIDTH — >. <— ^^ ^— UJ ^0.9 Q. </) OI mo.8 u. z ^0.7 1- o O > o ^ 0.5 >- o 3 0.4 m > ^0.3 o (r o ^0.2 o io., ■^^H;;^ . — /^ o^^ ^ ^ ^ ^^ / / y \a X y / ^ ^ / / / 6 ^ / 7 / f // < f/> \ ^-'^'fA y^. / / 1 / 1 'L f4 // 1 / / 1 1 /A '/ // // L ^i / 1 7 \\ 1 3 4 FREQUENCY 5 6 7 8 9 10 IN KILOMEGACYCLES PER SECOND 12 1 Fig. 1.3 — Theoretical group velocity vs. frequency curves for the 3" diameter ivaveguide used for the tests shown on Fig. 12. The vertical shaded area gives the bandwidth for the millimicrosecond pulses employed in that arrangement. 54 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 ordinary modulation methods. Darlington gives this formula: ^) ^^^^ iLLi/ Vnm where : jB = base bandwidth for 180° out of phase sidebands / = operating frequency (in same units as jB) X = wavelength in air L = waveguide length (in same units as X) Vnm = X/Xe Xc = cutoff wavelength for the mode involved With this equipment, the base bandwidth of the pulse is about 175 mc, and when/5 from the formula above is about equal to or less than this, pulse distortion should be observed. The following Table III gives fB calculated from this formula for the arrangement shown on Fig. 12. It is interesting to note that pulses in the TMu and TE31 modes, for which jB is less than the 175-mc pulse bandwidth, are broadened, but not badly distorted. For the higher modes, where jB is much less than 175 mc, broadening and severe distortion are evident. Another example is given in the next section. 9. DELAY DISTORTION EQUALIZATION If the distance which a pulse travels in a waveguide is increased, its delay distortion also increases. Since the group velocity at one edge of the band is different than at the other edge of the band, the amount by which the two edges get out of phase with each other increases with the total length of travel, causing increased distortion and pulse broaden- ing. The Darlington formula in the previous section shows that jB varies inversely as the square root of the length of travel. This efTect is shown on Fig. 14. In this arrangement the transmitter was connected to the end of a 3" diameter round waveguide 107 feet long through a small hole in the end plate. A mode filter was used so that only the TEoi mode would be transmitted in this Avaveguide. Through another small hole in the end plate polarized 90° from the first one, and rotated 90° around tlu^ plate, a directional coupler was connected as shown. The direct through guide of this directional coupler could be short cir- cuited with a waveguide shorting switch. Energy reflected from this fl WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 55 Table III - — Calculatee > Values of fB foe the Arrangement Shown in Fig. 1 2 Mode Number Mode Designation /B Megacycles Remarks 1 TEn 324.0 2 TMoi 237.7 3 TEn 174.9 4 TMu 124.1 5 TEoi 124.1 Not excited 6 TE31 105.2 7 TMoi 65.9 8 TE41 59.1 9 TEi, 58.6 Veiy weakly excited 10 TMoo 51.8 11 TM3: 21.3 Not observed 12 TE51 20.0 Not observed NUMBER OF R( 3UND TRIPS TAPERED DELAY DISTORTION EQUALIZER WAVEGUIDE SHORTING SWITCH 1/ 'M ^ te; >T0 RECEIVER NOT EQUALIZED (SWITCH CLOSED) EQUALIZED (SWITCH OPEN) TEqiIN 3 DIAM ROUND GUIDE (107 FT LONG) Fig. 14 — The left-hand series of pulses shows the build up of delay distortion with increasing number of round trips in a long waveguide. The right-hand series shows the im]irovement obtained with the tapered delay distortion equalizer shown at the right. 56 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 switch was then taken through the directional coupler to the receiver as shown by the output arrow. The series of pulses at the left-hand photograph of the oscilloscope traces was taken with this waveguide i shorting switch closed. The top pulse shows the direct leakage across the inside of the end plate before it has traveled through the 3" round guide. The next pulse is marked one round trip, having gone therefore 214 feet in the TEoi mode in the round waveguide. The successive pulses have traveled more round trips as shown by the number in the center between the two photographs. The effect of increased delay distortion broadening and distorting the pulse can be seen as the numbers increase. The values of fB from the Darlington formula in the previous section for these lengths are given in Table IV. It will be noticed that pulse broadening, and eventually severe dis- tortion, occurs as fB decreases much below the 175-mc pulse band- width. The effect is gradual, and not too bad a pulse shape is seen until fB is about half the pulse bandwidth, although broadening is very evident earlier. When the waveguide short-circuiting switch was opened so that the tapered delay distortion equalizer was used to reflect the energy, in- stead of the switch, the series of pulses at the right was observed on the indicator. It will be noted that there is much less distortion of these, pulses, particularly toward the bottom of the series. The ones at the top, have less distortion than would be expected, probably because of fre-, quency modulation of the injected pulse. The equalizer consists of a long gradually tapered section of waveguide which has its size reduced to a point beyond cutoff for the frequencies involved. Reflection takes place at the point of cutoff in this tapered guide. For the high frequency part of the pulse bandwidth, this point is farther away from the short- ing switch than for the low frequency part of the bandwidth. Conse- quently, the high frequency part of the pulse travels farther in one round trip into this tapered section and back than the low frequency part of Table IV — Values of fB from the Darlington Formula FOR the Arrangement Show^n in Fig. 14 li Round Trip Number JB Megacycles Round Trip Number fB Megacycles 1 2 3 4 5 185.8 131.4 107.3 92.9 83.1 6 7 8 9 10 75.8 70.2 65.7 61.9 58.7 j 1 1 WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 57 he pulse. This increased time of travel compensates for the shorter ime of travel of the high frequency edge of the band in the 3" round .vaveguide, so equalization takes place. Since this waveguide close to cutoff introduces considerable delay distortion by itself, the taper effect nust be made larger in order to secure the equalization. This can be ilone by making the taper sufficiently gradual. This type of equalizer ntroduces a rather high loss in the system. For this reason it might le used to predistort the signal at an early level in a repeater system, ilqualization by this method was suggested by J. R. Pierce. .0. MEASURING MODE CONVERSION FROM ISOLATED SOURCES I One of the important uses of this equipment has been for the meas- irement of mode conversion. W. D. Warters has cooperated in develop- ng techniques and carrying out such measurements. One of the prob- ems in the design of mode filters used for suppressing all modes except ;he circular electric ones in round multimode guides is mode conversion. Since these mode filters have circular symmetry, conversion can take alace only to circular electric modes of order higher than the TEoi mode. This conversion is, however, a troublesome one, since these higher Drder circular modes cannot be suppressed by the usual type of filter. An arrangement for measuring mode conversion at such mode filters rom the TEoi to the TE02 mode is being used with the short pulse equip- :nent. This employs a 400-foot long section of the b" diameter line. Be- ause the coupled- line transducer available had too high a loss to TE02 , a 3ombined TEoi — TE02 transducer was assembled. It uses one-half of :he round waveguide to couple to each mode. Fig. 15 shows this device. The use of this transducer and line is illustrated in Fig. 16. Pulses in :he TEoi mode are sent into the waveguide by the upper section of the transducer as shown. Some of the TEoi energy goes directly across to ohe TE02 transducer and appears as the outgoing pulse with a level down about 32 db. This is useful as a time reference in the system and s shown as the outgoing pulse in the photo of the oscilloscope trace ibove. The main energy in the TEoi mode propagates down the line as hown by dashed line 2, which is the path of this wave. Most of ohis energy goes all the way to the reflecting piston at the far end and ohen returns to the TE02 transducer where it gives a pulse which is narked TEoi round trip on the trace photograph above. Two thirds of ;he way from the sending end to the piston, the mode filter being meas- ired is inserted in the line. When the TEoi mode energy comes to this node filter, a small amount of it is converted to the TE02 mode. This 58 THE BELL SYSTEM TECHNICAL JOURNAL Fig. 15 — A special experimental transducer for injecting the TEoi mode and' receiving the converted TE02 mode in a 5" diameter waveguide. continues to the piston by path 4 (with dashed Hnes and crosses) and then returns and is received by the TE02 part of the transducer. This appears on the trace photo as the TE02 first conversion. When the main TEoi energy reflected by the piston comes back to the mode filter, conversion again takes place to TE02 • This is shown by path 3 hav- ing dashed lines and circles. This returns to the TE02 part of the trans- ducer and appears on the trace photo as the TE02 second conversion. In addition, a small amount of energy in the TE02 mode is generated by the TEoi upper part of the transducer. It is shown by path 5, having' OUTGOING PULSE TEoi ROUND TRIP TE02 SECOND CONVERSION TE02 FIRST CONVERSION TE02 ROUND TRIP ' MODE FILTER Fig. 16 — Trace photos and waveguide paths traveled when measuring TEoi, to TE02 mode conversion at a mode filter with the transducer shown on Fig. 15 All WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 59 jihort dashes. This goes down through the waveguide to the far end Ijiston and back, and is received by the TE02 transducer and shown as [he pulse marked TE02 round trip. The pulse marked TEoi round trip las a time separation from the outgoing pulse which is determined by ,he group velocity of TEoi waves going one round trip in the guide. The |rEo2 round trip pulse appears at a time corresponding to the group /elocity of the TE02 mode going one round trip in the guide. Spacing the node filter two-thirds of the way down produces the two conversion :)ulses equally spaced between these two as shown in Fig. 16. The first ponversion pulse appears at a time which is the sum of the time taken or the TEoi to go down to the filter and the TE02 to go from the filter uo the piston and back to the receiver. Because of the slower velocity bf the TE02 , this appears at the time shown, since it was in the TE02 node for a longer time than it was in the TEoi mode. The second con- [/ersion, which happened when TEoi came back to the mode filter, comes jiarlier in time than the first conversion, since the path for this signal ivas in the TEoi mode longer than it was in the TE02 mode. This arrange- ,nent gives very good time separation, and makes possible a measure- Inent of the amount of mode conversion taking place in the mode filters, viode conversion from TEoi to TE02 as low as 50 to 55 db down, can be neasured with this equipment. Randomly spaced single discontinuities in long waveguides can be ocated by this technique if they are separated far enough to give in- lividually resolved short pulses in the converted mode. Fig. 17 shows CONVERSION FIRST CONVERSION AT FAR END SECOND CONVERSION AT NEAR END SQUEEZED AT NEAR END SQUEEZED SECTION SECTION SQUEEZED SECTION TO RECEIVER TEJo — *- TEq, TE2, -• »- TE,o NEAR END 250 FT OF FAR END TRANSDUCER COUPLED LINE SQUEEZED 3"DIAM ROUND SQUEEZED TRANSDUCER SECTION GUIDE SECTION Fig. 17 — Arrangement used to explain the measurement and location of mode onversion from isolated sources. A deliberately squeezed section was placed t each end of the long waveguide, producing the pulses shown in the trace photo. 60 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 an arrangement having oval sections deliberately placed in the wave- ' guide in order to explain the method. Pure TEoi excitation is vised, and the converted TE21 mode observed with a coupled line transducer giv- ; ing an output for that mode alone. ; Let us consider first what would happen with the far-end squeezed; section alone, omitting the near-end squeezed section from considera- • tion. The injected TEoi mode signal would then travel down the 250 , feet of 3" diameter round waveguide to the far end with substantially, no mode conversion at the level being measured. At this point it goes through the squeezed section. Conversion now takes place from the TEou, mode to the TE21 mode. Both these modes after reflection from the piston travel back up the waveguide to the sending end. The group velocity of the TE21 mode is higher than the group velocity of the TEoi mode, so energy in these two modes separates, and if a coupling system were used to receive energy in both modes, two pulses would appear, with at time separation between them. In this case, since the receiver is con- nected to the line through the coupled line transducer which is responsive only to the TE21 mode, only one pulse is seen, that due to this mode alone. This is the center pulse in the trace photograph at the top of Fig. 17. If only one mode conversion point at the far end of the guide exists, only this one pulse is seen at the receiver. It would be spaced a distance away from the injected outgoing pulse that corresponds m:^ time to one trip of the TEoi mode down to the far end and one trip of || the TE21 mode from the far end back to the receiver. Now let us consider what would happen if the near-end squeezed sec- tion alone were present. When the TEqi wave passes the oval section! just beyond the coupled line transducer, conversion takes place, andi the energy travels down the line in both the TEoi and the TE21 modes,:; at a higher group velocity in the TE21 mode. These two signals are re- flected by the piston at the far end and return to the sending end. The TE21 signal comes through the coupled line transducer and appears as the pulse at the left of the photo shown on Fig. 17. Now the TEoi energy has lagged behind the TE21 energy, and when it gets back to the near- end squeezed section, a second mode conversion takes place, and TE21 mode energy is produced which comes through the coupled line trans-: ducer and appears at the receiver at the time of the right hand pulse. The spacing between these two pulses is equal to the difference in round trip times between the two modes. In general, for a single conversion source occurring at any point in the line, two pulses will appear on the scope. The spacing between these pulses corresponds to the difference in group velocity between the modes. WAVEGUIDE TESTING AVITH MILLIMICROSECOND PULSES 61 {from the point of the discontiimity down to the piston at the far end, land then back to the discontinuity. If the discontinuity is at the far lend, this time difference becomes zero, and a single pulse is seen. By i [making a measurement of the pulse spacing, the location of a single i icon version point can be determined. [ In the arrangement illustrated in Fig. 17, two isolated sources of j conversion existed. They were spaced far enough apart so that they \ were resolved by this equipment, and all three pulses were observed. The two outside pulses were due to the first conversion point. The center pulse was caused by the other squeeze, which was right at the reflecting |:)iston. If this conversion point had been located back some distance rom the piston, it would have produced two conversion pulses whose 'spacing could be used to determine the location of the conversion point. I The coupled-line transducers are calibrated for coupling loss by send- ng the pulse through a directional coupler into the branch normally ised for the output to the receiver. This gives a return loss from the lirectional coupler equal to twice the transducer loss plus the round rip line loss. 1. MEASURING DISTRIBUTED MODE CONVERSION IN LONG WAVEGUIDES ; Measurements of mode conversion from TEoi to a number of other nodes have been made with 5" diameter guides using this equipment, rhe arrangement of Fig. 18 was set up for this purpose. This is the same IS Fig. 17, except that a long taper was used at the input end of the 5" waveguide, and a movable piston installed at the remote end. One of the converted modes studied with this apparatus arrange- uent was the TMu mode, which is produced by bends in the guide, rhis mode has the same velocity in the waveguide as the TEoi mode. Therefore energy components converted at different points in the line tay in phase with the injected TEoi mode from which they are converted, rhere is never any time separation between these modes, and a single TO RECEIVER ■ I ■■'■ '■■■ ^ ^^i Si TEro— *TE^, COUPLED LINE -^^p^P, 5„ 0,^^^ MOVABLE TRANSDUCER TRANSDUCER HOLMDEL LINE PISTON FOR THE MODE BEING MEASURED Fig. 18 — Arrangement used for measuring mode conversion in the 5" diameter aveguides at Holmdel. 62 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 narrow pulse like the transmitted one is all that appears on the indicator oscilloscope. It is not possible from this to get any information about the location or extent of the conversion points in the line. Moving the far end piston does not change the relative phases of the modes, so no changes are seen in indicator pattern or pulse level as the piston is moved. For the Holmdel waveguides, which are about 500 feet long, the total round trip T]\In mode converted level varies from 32 to 36 db below the input TEoi mode level over a frequency range from 8,800 to 9,600 mc per second. All the other modes have velocities that are different than that of the TEoi mode. ^Vhen mode conversion takes place at many closely spaced points along the waveguide, the pulses from the various sources overlap, and phasing effects take place. In general, a filled-in pulse much longer than the injected one is observed. The maximum possible, but not necessary, pulse length is equal to the difference in time re- quired for the TEoi mode and the converted mode to travel the total waveguide length being observed. The phasing effects within the broad- ened pulse change its height and shape as a function of frequency and line length. Measurements of mode conversion from TEoi to TE31 in these wave- guides illustrate distributed sources and piston phasing effects. The TE3, mode has a group velocity 1.4 per cent slower than the TEoi mode. For a full round trip in the 500-foot lines, assuming conversion at the imput end, this causes a time separation of about two and one half pulse widths between these two modes. The received pulse is about two and a half times as long as the injected pulse, indicating rather closely spaced sources over the whole line length. For one far-end piston posi- tion, the received pattern is shown as the upper trace in Fig. 19. As the piston is moved, the center depressed part of the trace gradually ImK. 10 — Hocoivcd pulsr patterns willi llic .irraiijicnuMit of Fig. IS used for studying conversion to tlie Tlvn mode. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 63 rises until the pattern shown in the lower trace is seen. As the piston is moved farther in the same direction the trace gradually changes to have the appearance of the upper photo again. Moving the far-end piston changes the phase of energy on the return trip, and thus it can be made to add to, or nearly cancel out, conversion components that originated ahead of the piston. When the time separation becomes great enough to prevent overlapping in the pulse ^^^dth, phasing effects cannot take place, therefore, the beginning and end of the spread-out received pulse are not affected by moving the piston. Energy converted at the sending end of the guide travels the full round trip to the piston and back in the slower TE31 mode, and thus appears at the latest time, which is at the right-hand end of the received pulse. Conversion at the piston end returns at the center of the pulse, and conversion on the return trip comes at earlier times, at the left-hand part of the pulse. The TEoi mode has less loss in the guide than the TE31 mode. Since the energy in the earlier part of the received pulse spent a greater part of the trip in the lower loss TEoi mode before conversion, the output is higher here, and slopes off toward the right, where the later returning energy has gone for a longer distance in the higher loss mode. The pulse height at the maximum shows the converted energy from that part of the line to be between 30 and 35 db below the incident TEoi energy level over the measured band\\ddth. Measurements of mode conversion from TEoi to TE21 in these wave- guides show these same effects, and also a phasing effect as a function of frequency. The TE21 mode has a group velocity 2.4 per cent faster than the TEoi mode. For a full round trip in the guides, this is a time separation of about four pulse mdths between the modes. At one fre- quency and one far-end piston position, the TE21 response shown as the top trace of Fig. 20 was obtained. Moving the far-end piston gradually changed this to the second trace from the top, and further piston mo- tion changed it back again. This is the same kind of piston phasing effect observed in the TE31 mode conversion studies. The irregular top of this broadened pulse indicates fewer conversion points than for the TE31 mode, or phasing effects along the guide length. Since the TE21 mode has a higher group velocity' than the TEoi mode, energy converted at the beginning of the guide returns at the earlier or left-hand part of the pulse, and conversions on the return trip, having traveled longer in the slower TEoi mode, are on the right-hand side of the pulse. This is just the reverse of the situation for the TE31 mode. Since the loss in the TE21 mode is higher than in the TEoi mode, the right side of this broad- ened pulse is higher than the left side, as the energy in the left side has 64 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 gone further in the higher loss TE21 mode. Conversions from the piston end of the guide return in the center of the pulse, and only in this re- gion do piston phasing effects appear. As the frequency is changed the ' pattern changes, until it reaches the extreme shape shown in the next- to-the-bottom trace, with this narrower pulse coming at a time corre- sponding to the center of the broadened pulse at the top. Further fre- quency change in the same direction returns the shape to that of the top traces. At the frequency giving the received pulse shown on the next-to-the-bottom trace, moving the far-end piston causes a gradual change to the shape shown on the lowest trace. This makes it appear as if the mode conversion were coming almost entirely from the part of the guide near the piston end at this frequency. The upper traces appear to show that more energy is converted at the transducer end of the waveguide at that frequency. It would seem that at certain frequencies some phase cancellation is taking place between conversion points spaced closely enough to overlap within the pulse width . At frequencies between the ones giving traces like this, the appearance is more like that shown for the TE31 mode on Fig. 19 except for the slope across the top of the pulse being reversed. The highest part of this TEoi pulse is Fiff. 20 — Received pulse patterns witli the urrangemeiit of Fig. 18 used for studying conversion to the TE21 mode. WAVEGUIDE TESTING WITH MILLIMICKOSECOND PULSES 65 24 to 27 db below the injected TEoi pulse level for the 5" diameter Holmdel waveguides. 12. CONCLUDING REMARKS The high resolution obtainable with this millimicrosecond pulse equipment provides information difficult to obtain by any other means. These examples of its use in waveguide investigations indicate the possibilities of the method in research, design and testing procedures. It is being used for many other similar purposes in addition to the illus- tratio)is given here, and no doubt many more uses will be found for such short pulses in the future. REFERENCES 1. S. E. Miller and A. C. Beck, Low-loss Waveguide Transmission, Proc. I.R.E., 41, pp. 348-358, March, 1953. 2. S. E. Miller, Waveguide As a Communication Medium, B. S. T. J., 33, pp. 1209- 1265, Nov., 1954. 3. C. C. Cutler, The Regenerative Pulse Generator, Proc. I.R.E., 43, pp. 140- 148, Feb., 1955. 4. S. E. Miller, Coupled WaveTheory and Waveguide Applications, B. S. T. J., 33, pp. 661-719, May, 1954. Experiments on the Regeneration of Binary Microwave Pulses By O. E. DeLANGE (Manuscript received September 7, 1955) A sifnple device has been produced for regenerating binary pulses directly at microwave frequencies. To determine the capabilities of such devices one of them was included in a circidating test loop in which pidse groups were passed through the device a large number of titnes. Residts indicate that even in the presence of serious noise and bandwidth limitations pidses can be regenerated many times and still shotv no noticeable deterioration. Pic- tures of circulated pidses are included which illustrate performance of the regenerator. INTRODUCTION The chief advantage of a transmission system employing Ijinary pulses resides in the possibility of regenerating such pulses at intervals along the route of transmission to prevent the accumulation of distortion due to noise, bandwidth limitations and other effects. This makes it possible to take the total allowable deterioration of signal in each section of a long relay system rather than having to make each link sufficiently good to prevent total accumulated distortion from becoming excessive. This has been pointed out by a number of writers. i-- W. M. GoodalP has shown the feasibility of transmitting television signals in binary form. Such transmission reciuires a considerable amount of bandwidth; a seven digit system, for example, would require trans- mission of seventy million pulses per second. This need for wide bands makes the microwave range an attractive one in which to work. S. E. Miller* has pointed out that a binary system employing regeneration might prove to be especially advantageous in waveguide transmission. 1 B. M. Oliver, J. R. Pierce and, C. E. Shannon, The Pliilosophv of PCM, Proc. I. R.E., Nov., 1948. '^ L. A. Meacham and Iv Peterson, An Experimental Multichannel Pulse Code Modulation System of Toll Quality, B. S. T. J., Jan. 1948. ' W. M. Goodall, Television l)y Pulse Code Modulation, B. S. T. J., Jan., 1951. * S. E. Miller, Waveguide as a Communication Medium, B. S. T. J., Nov., 1954. 67 68 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INPUT FILTER AUTOMATIC GAIN CONTROL REGENERATOR DETECTOR TIMING WAVE GENERATOR FILTER OUTPUT Fig. 1 — A typical regenerative repeater shown in block form. That the Bell System is interested in the long-distance transmission of television and other broad-band signals is evident from the number of miles of such broad-band circuits, both coaxial cable and microwave radio, ^ now in service. These circuits provide high-grade transmission because each repeater was designed to have a very fiat frequency charac- teristic and linear phase over a considerable bandwidth. Furthermore, these characteristics are very carefully maintained. For a binary pulse system employing regeneration the requirements on flatness of band and linearity of phase can be relaxed to a considerable degree. The compo- nents for such a system should, therefore, be simpler and less expensive to build and maintain. Reduced maintenance costs might well prove to be the chief virtue of the binary system. Since the chief advantage of a binary system lies in the possibility of regeneration it is obvious that a very important part of such a system is the regenerative repeater employed. Fig. 1 shows in block form a typical broad-band, microwave repeater. Here the input, which might come from either a radio antenna or from a waveguide, is first passed through a proper microwave filter then amplified, probably by a traveling-wave amplifier. The amplified pulses of energy are regenerated, filtered, am- plified and sent on to the next repeater. The experiment to be described here deals primarily with the block labeled "Regenerator" on Fig. 1. In these first experiments one of our main objectives was to keep the repeater as simple as possible. This suggests regeneration of pulses directly at microwave frequency, which for this experiment was chosen to be 4 kmc. It was suggested by J. R. Pierce and W. D. Lewis, both of Bell Telephone Laboratories, that further simplification might be made possible by accepting only partial instead of complete regeneration. This suggestion was adopted. For the case of complete regeneration each incoming pulse inaugurates a new pulse, perfect in shape and correctly timed to be sent on to the 'A. A. Roetken, K. D. Smith and R. W. Friis, The TD-2 System, B. S. T. J., Oct., 1951, Part II. REGENERATION OF BINARY MICROWAVE PULSES 69 next repeater. Thus noise and other disturbing effects are completely eliminated and the output of each repeater is identical to the original signal which entered the system. For the case of partial regeneration incoming pulses are retimed and reshaped only as well as is possible with simple equipment. Obviously the difference between complete and partial . regeneration is one of degree. One object of the experiment was to determine how well such a partial regenerator would function and what price must be paid for employing partial instead of complete regeneration. The regenerator developed consists simply of a waveguide hybrid junction with a silicon crystal diode in each side arm. It appears to meet the requirement of simplicity in that it combines the functions of amplitude slicing and pulse retiming in one unit. A detailed description of this unit will be given later. Al- though the purpose of this experiment was to determine what could be accomplished in a very simple repeater we must keep in mind that superior performance would be obtained from a regenerator which ap- proached more nearly the ideal. For some applications the better re- generator might result in a more economical system even though the regenerator itself might be more complicated and more expensive to produce. METHOD OF TESTING The regeneration of pulses consists of two functions. The first function is that of removing amplitude distortions, the second is that of restoring each pulse to its proper time. The retiming problem divides into two [parts the first of which is the actual retiming process and the second ! that of obtaining the proper timing pulses with which to perform this lifunction. In a practical commercial system timing information at a [repeater would probably be derived from the incoming signal pulses. There are a number of problems involved in this recovery of timing pulses. These are being studied at the present time but were avoided in the experiment described here by deriving such information from the local synchronizing gear. Since the device we are dealing with only partially regenerates pulses it is not enough to study the performance of a single unit — we should •like to have a large number operating in tandem so that we can observe 'what happens to pulses as they pass through one after another of these Tegenerators. To avoid the necessity of building a large number of units the pulse circulating technique of simulating a chain of repeaters was j employed. Fig. 2 shows this circulating loop in block form. 70 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 HYBRID JUNCTION ' NO. 3 CW OSCILLATOR (4 KMC) TRAVELING WAVE AMPLIFIER (NOISE GENERATOR) Fig. 2 — The circulating loop. To provide RF test pulses for this loop the output of a 4 kmc, cw oscillator is gated by baseband pulse groups in a microwave gate or modulator. The resultant microwa\-e pulses are fed into the loop (heavy line) through hybrid junction No. 1. They are then amplified by a trav- eling-wave amplifier the output of which is coupled to the pulse regen- erator through another hybrid junction (No. 2). The purpose of this hybrid is to provide a position for monitoring the input to the regen- erator. A monitoring position at the output of the regenerator is pro- vided by a third hybrid, the main output of which feeds a considerable length of waveguide which provides the necessary loop delay. At the far end of the waveguide another hybrid (No. 4) makes it possible to feed noise, which is derived from a traveling-wave amplifier, into the loop. The combined output after passing through a band pass filter is ampli- REGENEKATION OF BINARY MICROWAVE PULSES 71 fied by another traveling-wave amplifier and fed back into the loop in- put thus completing the circuit. The synchronizing equipment starts out with an oscillator going at approximately 78 kc. A pulse generator is locked in step with this os- cillator. The output of the pulser is a negative 3 microsecond pulse as shown in Fig. 3A. After being amplified to a level of about 75 volts this pulse is applied to the helix of the first traveling-wave tube to re- I duce the gain of this tube during the 3-microsecond interval. Out of each 12.8/xsec interval pulses are allowed to circulate for O.S/xsec but are blocked I for the remaining 3Msec thus allowing the loop to return to the quiescent i condition once during each period as shown on Figs. 3A and 3C. The S^sec pulse also synchronizes a short-pulse generator. This unit delivers pulses which are about 25 millimicroseconds long at the base and spaced by 12.8/isec, i.e., Avith a repetition frequency of 78 kc. See Fig. 3B. In order to simulate a PCM system it was decided to circulate pulse CIRCULATING INTERVAL 9.8/ZS QUENCHING INTERVAL -3//S-*| (A) GATING CYCLE (B) SHORT SYNCHRONIZING PULSES --24 GROUPS OF PULSES (C) CIRCULATING PULSE GROUPS GROUP GROUP GROUP 1 2 3 lOOMyUS ^ k ^^-o.4;uS-^^ I (D) PULSE GROUPS (EXPANDED) ■ ' |300M/US| I I I (E) TIMING WAVE (40MC) EXPANDED TIME Fig. 3 — Timing events in the circulating loop. 72 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 groups rather than individual pulses through the system. These were derived from the pulse group generator which is capable of delivering any number up to 5 pulses for each short input pulse. These pulses are about 15 milli-microseconds long at the base and spaced 25 milli-micro- seconds apart. The amplitude of each of these pulses can be adjusted independently to any value from zero to full amplitude making it pos- sible to set up any combination of the five pulses. These are the pulses which are used to gate, or modulate, the output of the 4-kmc oscillator. The total delay around the waveguide loop including TW tubes, etc.,' was 0.4)usec or 400 milli-microseconds. This was sufficient to allow time between pulse groups and yet short enough that groups could circulate 24 times in the available 9.8jLtsec interval. This can be seen from Figs. 3C and 3D. The latter figure shows an expanded view of circulating pulse groups. The pulses in Group 1 are inserted into the loop at the beginning of each gating cycle, the remaining groups result from circu- lation around the loop. When all five pulses are present in the pulse groups the pulse repeti- tion frequency is 40 mc. (Pulse interval 25 milli-microseconds). For this condition timing pulses should be supplied to the regenerator at the rate of 40 million per second. These pulses are supplied continuously and not in groups as is the case with the circulating pulses. See Fig. BE. In order to maintain time coincidence between the circulating pulses and the tim- ing pulses the delay around the loop must be adjusted to be an exact multiple of the pulse spacing. In this experiment the loop delay is equal to 16-pulse intervals. Since timing pulses are obtained by harmonic generation from the quenching frequency as will be discussed later this frequency must be an exact submultiple of pulse repetition frequency. In this experiment the ratio is 512 to 1. Although the above discussion is based on a five-pulse group and 40-mc repetition frequency it turned out that for most of the experi- ments described here it was preferable to drop out every other pulse, leaving three to a group and resulting in a 20-mc repetition frequency. The one exception to this is the limited-band-width experiment which will be described later. - For all of the experiments described here timing pulses were derived from the 78-kc quenching frequency by harmonic generation. A pulse with a width of 25 milli-microseconds and with a 78-kc repetition fre- quency as shown in Fig. 3B supplied the input to the timing wave gen- erator. This generator consists of several stages of limiting amplifiers all tuned to 20 mc, followed by a locked-in 20-mc oscillator. The output of the amplifier consists of a train of 20-mc sine waves with constant ampli- til REGENERATION OF BINARY MICROWAVE PULSES 73 tude for most of the 12.8Msec period but falling off somewhat at the end of the period. This-train locks in the oscillator which oscillates at a con- stant amplitude over the whole period and at a frequency of 20 mc. Timing pulses obtained from the cathode circuit of the oscillator tube pro^'ided the timing waves for most of the experiments. For the experi- ment where a 40-mc timing wave was required it was obtained from the, 20 mc train by means of a frequency doubler. For this case it is necessary for the output of the timing wave generator to remain constant in ampli- tude and fixed in phase for the 512-pulse interval between synchronizing pulses. In spite of the stringent requirements placed upon the timing equip- ment it functioned well and maintained synchronism over adequately long periods of time without adjustment. PERFORMANCE OF REGENERATOR Performance of the regenerator under various conditions is recorded on the accompanying illustrations of recovered pulse envelopes. The first experiment was to determine the effects of disturbances which arise at only one point in a system. Such effects were simulated by adding disturbances along with the group of pulses as they were fed into the circulating loop from the modulator. This is equivalent to having them occur at only the first repeater of the chain. Some of the first experiments also involved the use of extraneous pulses to represent noise or distortion since these pulses could be syn- chronized and thus studied more readily than could random effects. In , Fig. 4A the first pulse at the left represents a desired digit pulse with ' its amplitude increased by a burst of noise, the second pulse represents ' a clean digit pulse, and the third pulse a burst of noise. This group is at 1 the input to the regenerator. Fig. 4B shows the same group of pulses ' after traversing the regenerator once. The pulses are seen to be shortened due to the gating, or retiming, action. There is also seen to be some ampli- tude correction, i.e. the two desired pulses are of more nearly the same j amplitude and the undesired pulse has been reduced in relative ampli- tude. After a few trips through the regenerator the pulse group was rendered practically perfect and remained so for the rest of the twenty- four trips around the loop. Fig. 4C shows the group after 24 trips. In 'another experiment pulses were circulated for 100 trips without deteri- oration. Nothing was found to indicate that regeneration could not be repeated indefinitely. Figs. 5 A and 5B represent the same conditions as those of 4 A and 4B 74 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 4 — Effect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, original signal. B — Output of regenerator, first trip. C — Output of regenerator, 24th trip. Fig. 5 — l']ffect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, first four groups. B — Output of regenerator, first four groups. C — Output of regenerator, increased input level. REGENERATION OF BINARY MICROWAVE PULSES 75 Fig. 6 — Effect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, original signal. B — ^ Output of regenerator, first trip. C • — Oi^tput of regenerator, 24th trip. except that the oscilloscope sweep has been contracted in order to show the progressive effects produced by repeated passage of the signal through the regenerator. Fig. 5B shows that after the pulses have passed through the regenerator only twice all visible effects of the disturbances have been removed. Fig. 5C shows the effect of simply increasing the RF pulse input to the regenerator by approximately 4 db. The small "noise" pulse which in the previous case was quickly dropped out because of being below the slicing level has now come up above the slicing level and so builds up to full amplitude after only a few trips through the regenerator. Note that in the cases shown in Figs. 4 and 5 discrimination against unwanted pulses has been purely on an amplitude basis since the gate has been unblocked to pulses with amplitudes above the slicing level whenever one of these distiu'bing pulses was present. For Fig. 6A conditions are the same as for Fig. 4A except that an ad- ditional pulse has been added to simulate intersymbol noise or inter- ference. Fig. 6B indicates that after only one trip through the regenerator the effect of the added pulse is very small. After a few trips the effect is completely eliminated leaving a practically perfect group which con- tinues on for 24 trips as shown by Fig. 6C. For the intersymbol pulse, discrimination is on a time basis since this interference occurs at a time 76 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 7 — Effect of regenerating in amplitude without retiming. A — Outputof regenerator, no timing, firt trip. B — Output of regenerator, no timing, 10th trip. Output of regenerator, no timing, 23rd trip. when no gating pulse is present and hence finds the gate blocked regard- less of amplitude. To show the need for retiming the pictures shown on Figs. 7 and 8 were taken. These were taken with the amplitude slicer in operation but with the pulses not being retimed. Figs. 7A, 7B and 7C, respectively, show the output of the slicer for the first, tenth and twenty-third trips. After ten trips, there is noticeable time jitter caused by residual noise in the system; after 23 trips this jitter has become severe though pulses are still recognizable. It should be pointed out that for this experiment no noise was purposely added to the system and hence the signal-to- noise ratio was much better than that which would probably be encoun- tered in an operating system. For such a system we would expect time jitter effects to build up much more rapidly. For Fig. 8 conditions are the same as for Fig. 7 except that the pulse spacing is decreased by the addition of an extra pulse at the input. Now, after ten trips, time jitter is bad and after 23 trips the pulse group has become little more than a smear. This increased distortion is probably due to the fact that less jitter is now required to cause overlap of pulses. There may also be some effects due to change of duty cycle. For Fig. 9 there was neither slicing nor retiming of pulses. Here, pulse groups deteriorate very rapidly to nothing more than blobs of energy. Note that there is an increase of i REGENERATION OF BINARY MICROWAVE PULSES 77 Fig. 8 — ■ Effect of regenerating in amplitude without retiming. A — Output of regenerator, no timing, first trip. B — Output of regenerator, no timing, 10th trip. C — Output of regenerator, no timing, 23rd trip. Fig. 9 — Pulses circulating through the loop without regeneration. A — Origi- nal input. B — 4th trip without regeneration. C — 20th to 24th trip without re- generation. '8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 iWWWMMMWWIWWMilJflM ^ I I . rlil l lT- i \m....: i n i T i inr- IH. Fig. 10 — The regeneration of band-limited pulses. A — Input to regenerator, first two groups. B — Output of regenerator, first two groups. C — Output of regenerator, 24th trip. amplitude with each trip around the loop indicating that loop gain was slightly greater than unity. Without the sheer it is difficult to set the gain to exactly unity and the amplitude tends to either increase or de- : crease depending upon whether the gain is greater or less than unity. Results indicated by the pictures of Fig 9 are possibly not typical of a properly functioning system but do show what happened in this par- . ticular sj^stem when regeneration was dispensed with. Another important function of regeneration is that of overcoming . band-limiting effects. Figs. 10 and 11 show what can be accomplished. . For this experiment the pulse groups inserted into the loop were as shown i| at the left in Fig. lOA. These pulses were 15 milli-microseconds wide at the base and spaced by 25 milli-microseconds which corresponds to a j repetition frequency of 40 mc. After passing through a band-pass filter these pulses were distorted to the extent shown at the right in Fig. lOA. From the characteristic of the filter, as shown on Fig. 12, it is seen that the bandwidth employed is not very different from the theoretical min- imum required for double sideband transmission. This minimum char- acteristic is shown by the dashed lines on Fig. 12. Fig. lOB shows that at the output of the regenerator the effects of band limiting have been removed. This is borne out by Fig. IOC which shows that after 24 trips the code group was still practically perfect. It should l)e pointed out that the pulses traversed the filter once for each trip around the loop, REGENERATION OF BINARY MICROWAVE PULSES 79 Fig. 11 — The regeneration of band-limited pulses. A — Input to regenerator, first two groups. B — Output of regenerator, first two groups. C — Output of re- generator, 24th trip. that is for each trip the input to the regenerator was as shown at the right of Fig. lOA and the output as shown by Fig. lOB. It is important to note that Fig. 12 represents the frequency characteristic of a single hnk of the simulated system. The pictures of Fig. 11 show the same experi- ment but this time with a different code group. Any code group which we could set up with our five digit pulses was transmitted equally well. In order to determine the breaking point of the experimental system, broad-band noise obtained from a traveling-wave amplifier was added into the system as shown on Fig. 2. The breaking point of the system is the noise level which is just sufficient to start producing errors at the output of the system.* The noise is seen to be band-limited in exactly the same way as the signal. With the system adjusted to operate properly the level of added noise was increased to the point where errors became barely discernible after 24 trips around the loop. Noise level was now reduced slightly (no errors discernible) and the ratio of rms signal to rms noise measured. Fig. 13A shows the input to the regenerator for the 23rd and 24th trips with this amount of noise added. Note that the noise has * The type of noise employed has a Gaussian amplitude distribution and there- fore there was actually no definite breaking point — the rate at which errors Oc- curred increased continuously as noise amplitude was increased. The breaking point was taken as the noise level at which errors became barely discernible on the viewing oscilloscope. More accurate measurements made in other experiments indicate that this is a fairly satisfactory criterion. 80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 28 24 IT) aJ2o 03 O 16 to If) g 12 a. UJ 5 8 ll 1 — 1 \ i A \ ^ 1 1 1 1 1 / < \ 1 1 1 / / \ 1 / < V h* --20 M 1 --20MC *] 1/ v. 1 / 1 ■~ TX^ ^ ■rrD*^ /J 3950 3960 3970 3980 3990 4000 4010 4020 4030 4040 FREQUENCY IN MEGACYCLES PER SECOND Fig. 12 — Characteristics of the band-pass microwave filter. m % I JYYYYYYTin Fig. 13. — The regeneration of pulses in the presence of broad-hand, random noise added at each repeater. A — Ini)ut to regenerator, 23rd and 24th trijis, broad-band noise added. B — Ini)ut to regenerator, 23rd and 24th trips, no added noise. C — 20-mc timing wave. \ KEGENERATION OF BINARY MICROWAVE PULSES 81 Fig. 14 — The regeneration of pulses in the presence of interference occurring at each repeater. A — Original signal with added moduhited carrier interference. B — Input to regenerator, 24th trip, niochilatod carrier interference. C — Output of regenerator, 24th trip, modulated carrier interference. produced a considerable broadening of the oscilloscope trace. Fig. 13B shows the same pulse groups with no added noise. These photographs are included to give some idea as to how bad the noise was at the l;)reaking point of the system. Of course maximum noise peaks occur rather infre- quently and do not show on the photograph. At the output of the re- generator effects due to noise were barely discernible. This output looked so much like that shown at Fig. 14C that no separate photograph is shown for it. Figs. 14A, 14B and 14C show the effects of a different type of inter- ference upon the system. This disturbance was produced by adding into the system a carrier of exactly the same frequency as the signal carrier (4 kmc) but modulated by a 14-mc wave, a frequency in the same order as the pulse rate. Here again the level of the interference was adjusted to be just below the l)reaking point of the system. A comparison between Figs. 14B and 14C gives convincing evidence that the regenerator has substantially restored the waveform. For the case of the interfering signal a ratio of signal to interference of 10 db on a peak-to-peak basis was measured when the interference was just below the breaking point of the system. This, of course, is 4 db above the theoretical value for a perfect regenerator. For the case of 82 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 broad-band random noise an rms signal to noise ratio of 20 dl) was meas- ured.* This compares Avith a ratio of 18 db as measured by Messrs. Meacham and Peterson for a system employing complete regeneration and a single repeater, f Recently, A. F. Dietrich repeated the circulating loop experiment at a radio frequency of 11 kmc. His determinations of required signal-to- noise ratios are substantially the same as those reported here. From the various experiments we conclude that for a long chain of properly func- tioning regenerative repeaters of i-he type discussed here practically perfect transmission is obtained as long as the signal-to-noise ratio at the input to each repeater is 20 db or better on an rms basis. In an operat- ing system it might be desirable to increase this ratio to 23 db to take care of deficiencies in automatic gain controls, power changes, etc. From the experiments we also conclude that the price we pay for using partial instead of complete regeneration is about 3 to 4 db increase in the required signal-to-noise ratio. In a radio system which provides a fading margin this penalty would be less since the probability that two or more adjacent links will reach maximum fades simultaneously is very ' small. Under these conditions only one repeater at a time would be near the breaking point and the system would behave much as though the repeater provided complete regeneration. TIMING Although we have considered the problem of retiming of signal pulses up to now we have not discussed the problem of obtaining the necessary ' timing pulses to perform this function, but have simpl}^ assumed that a source of such pulses was available. As w^as mentioned earlier timing I pulses would probably be derived from the signal pulses in a practical »^ system. These pulses would be fed into some narrow band amplifier tuned to pulse repetition frequency. The output of this circuit could be made to be a sine wave at repetition frequency if gaps between the input pulses were not too great. Timing pulses could be derived from this sine wave. This timing equipment could be similar to that used in these ex- periments and described earlier. Further study of the problems of ob- taining timing information is being made. * For Gaussian noise it is not possible to specif.y a theoretical value of minimum S/N ratio without specifying the tolerable percentage of errors. For the number of errors detectable on the oscilloscope it seems rasonable to assume a 12 db peak factor for the noise. The peak factor for the signal is 3 db. The 6 db peak S/N which would be required for an ideal regenerator then becomes 15 db on an rms basis. t L. A. Meacham and E. Peterson, B. S. T. J., p. 43, Jan., 1948. " KEGENERATION OF BINARY MICROWAVE PULSES 83 ' GATING PULSE INPUT OUTPUT Fig. 15A — Low-frequency equivalent of the partial regenerator. DESCRIPTION OF REGENERATOR This device regenerates pulses by performing on them the operations of ''slicing" and retiming. An ideal slicer is a device with an input-output characteristics such as shown by the dashed lines of Fig. 15C. It is seen that for all input levels below the so-called slicing level transmission through the device is zero but that for all amplitudes greater than this value the output level is finite and constant. Thus, all input voltages which are less than the slic- ing level have no effect upon the output whereas all input voltages greater than the slicing level produce the same amplitude of output. Normally conditions are adjusted so that the slicing level is at one-half INPUT LEVEL Fig. 15B — Characteristics of the separate branches with ditterential bias. 84 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INPUT LEVEL Fig. 15C — Resultant output with differential bias. BRANCH 2 BRANCH 1 RESULTANT INPUT LEVEL Fig. 15D — Characteristics of the separate branches and resultant output with equal biases. of peak pulse amplitude — then at the output of the slicer there will be no effect whatsoever from disturbances unless these disturbances exceed half of the pulse amplitude. It is this slicing action which removes the amplitude effects of noise. Time jitter effects are removed by retiming, i.e., the device is made to have high loss regardless of input level except at those times when a gating pulse is present. Fig. 15A shows schematically a low-frequency equivalent of the re- generator used in these experiments. Here an input line divides into two identical branches isolated from each other and each with a diode shunted across it. The outputs of the two branches are recombined through neces- sary isolators to form a single output. The phase of one branch is re- versed before recombination, so that the final output is the difference between the two individual outputs. Fig. 15B shows the input-output characteristics of the two branches when the diodes are biased back to be non-conducting by means of bias voltages Vi and V2 respectively. For low levels the input-output char- acteristic of both branches will be linear and have a 45° slope. As soon REGENEKATION OF BINARY MICROWAVE PULSES 85 as the input voltage in a branch reaches a vakie equal to that of the back bias the diode will start to conduct, thus absorbing power and decrease the slope of the characteristic. The output of Branch 1 starts to flatten off when the input reaches the value Vi , while the output of Branch 2 does not flatten until the input reaches the value V2 . The combined output, which is equal to the differences of the two branch outputs, is then that shown by the solid line of Fig. 15C and is seen to have a transi- tion region between a low output and a high output level. If the two branches are accurately balanced and if the signal voltage is large com- pared to the differential bias V2 — Vi the transition becomes sharp and the device is a good slicer. If the two diodes are equally biased as shown on Fig. 15D the outputs of the two branches should be nearly equal regardless of input and the total output, which is the difference between the two branch outputs, will always be small. Fig. 16 shows a microwave equivalent of the circuit of Fig. 15A. In the microwave structure lengths of wave-guide replace the wire lines and branching, recombining and isolation are accomplished by means of hybrid junctions. The hybrid shown here is of the type known as the lA junction. Fig. 17 shows another equivalent microwave structure employing only one hybrid. This is the type used in the experiments described here. The [output consists of the combined energies reflected from the two side jarms of the junction. With the junction connected as shown phase rela- Itionships are such that the output is the difference between the reflec- GATING PULSE ^(— r-V\^^^ RF INPUT ARM PROBE TERMINATION I ARM 4 I— vw-^ Fig. 16 — Microwave regenerator. 86 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 tions from the two side arms so that when conditions in the two arms are identical there is no output. The crystal diodes coupled to the side arms are equivalent to those shunted across the two lines of Fig. 15A. Fig. 18, which is a plot of the measured input-output characteristic of the regenerator used in the loop test, shows how the device acts as a combined sheer and retimer. Curve A, ol)tained with equal biases on the two diodes, is the characteristic with no gating pulse applied i.e. the diodes are normally biased in this manner. It is seen that this condition produces the maximum of loss through the device. By shifting one diode bias so as to produce a differential of 0.5 volt the characteristic changes to that of Curve B. This differential bias can be supplied by the timing pulse in such a way that this pulse shifts the characteristic from that shown at A to that shown at B thus decreasing the loss through the de- vice by some 12 to 15 db during the time the pulse is present. In this way the regenerator is made to act as a gate — though not an ideal one. We see from curve B that with the differential bias the device has the characteristic of a slicer — though again not ideal. For lower levels of input there is a region over which the input-output characteristic is square law with a one db change of input producing a two db change of output. This region is followed by another in which limiting is fairly pronounced. At the 8-db input level, which is the point at which limiting sets in, the loss through the regenerator was measured to be approxi- mately 12 db. The characteristic shown was found to be reproducible both in these experiments at 4 kmc and in those bj'- A. F. Dietrich at 11 kmc. For a perfect slicer only an infinitesimal change of input level is re- GATING PULSE ■AAV-i_ ARM 2 RF OUTPUT Fig, 17 — Microwave regenerator employing a single hybrid junction. REGENERATION OF BINARY MICROWAVE PULSES 87 ID m o LU a D 3 o -10 -12 -14 -16 -18 -20 -22 -24 V, = 0.5 V2 = <-- 12 DB LOSS i>— <! P'< JH " ^ /I ^ 1 1 ) ( 1 ( r 1 6DB 1 1 r -6DB- 1 1 .--J (B) / V, = V2 A f / / Y / / / /I 1 (A)/ / 1/ / ^ y K / 6 8 10 12 INPUT LEVEL IN DECIBELS 14 16 18 Fig. 18 — Static characteristics of the regenerator employed in these experiments. f}uired to change the output from zero to maximum. The input level at which this transition takes place is the slicing level and has a very defi- nite value. For a characteristic such as that shown on Fig. 18 this point is not at all definite and the question arises as to how one determines the slicing level for such a device. Obviously this point should be somewhere on the portion of the characteristic where expansion takes place. In the case of the circulating loop the slicing level is the level for which total gain around the loop is exactly etiual to unity. Why this is so can be seen from Fig. 19 which is a plot of gain \'ersus input level for a repeater containing a sheer with a characteristic as shown by curve B of Fig. 18. Amplifiers are necessary in the loop to make up for loss through the re- generator and other components. For Fig. 11) we assume that these amplifiers have been adjusted so that gain around the loop is exactly unity for an input pulse having a peak amplitude corresponding to the 88 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 -3 -2-1 1 2 3 4 5 INPUT LEVEL IN DECIBELS ABOVE SLICING LEVEL Fig 19 — Gain characteristics of u repeater providing partial regeneration. point F' of Fig. 18. On Fig. 19 all other levels are shown in reference to this unity-gain value. From Fig. 19 it is obvious that a pulse which starts out in the loop with a peak amplitude exactly equal to the reference, or slicing level, will continue to circulate without change of amplitude since for this level there is unity gain around the loop. A pulse with amplitude greater than the slicing level will have its amplitude increased by each passage through a regenerator until it eventually reaches a value of +6 db. It will continue to circulate at this amplitude, for here also the gain around the loop isVmity.* Any pulse with peak amplitude less than the reference level will have its amplitude decreased by successive trips through the regenerator and eventually go to zero. We also see that the greater the departure of the amplitude of a pulse from the slicing level the more effect the regenerator has upon it. This means that the device acts much more powerfully on low level noise than on noise with pulse peaks near the slicing level. As examples consider first the case of noise peaks only 1 db below slicing level at the input (peak S/N = 7 db). At this level there is a 1 db loss through the repeater so that at the output the noise peaks will be 2 db below reference to give a *S/A^ ratio of 8 db. Next * Note that llic ^-fi-dl) level is at a point of stable equilibrium whereas at the slicing level C(iuilil)rium is unstable. REGENERATION OF BINARY MICROWAVE PULSES 89 consider noise with a peak level 5 db below slicing level (S/N =11 db) at the input. The loss at this level is 5 db resulting in a noise level 10 db below reference to give a S/N ratio of 16 db. We see that a 4 db improve- ment in S/N ratio at the input results in an 8 db improvement in this ratio at the output. Everything which was said above concerning the circulating loop ap- plies equally to a chain of identical repeaters. To set the effective slicing level at half amplitude at each repeater in a chain one would first find two points on the sheer characteristics such as P and P' of Fig. 18. The point P should be in the region of expansion and P' in the limiting region. Also the points should be so chosen that a 6 db increase of input from that at point P results in a 6 db increase in output at the point P'. If now at each repeater we adjust pulse peak amplitude at the sheer input to a value corresponding to that at point P' we will have unity gain from one repeater to the next at levels corresponding to pulse peaks. We will also have unity gain at levels corresponding to one half of pulse amplitude. The effective slicing level is thus set at half amplitude. Ob- viously the procedure for setting the slicing level at some value other than half amplitude would be practically the same. It should be pointed out that although half amplitude is the preferred slicing level for base- band pulses this is not the case for carrier pulses. W. R. Bennett of Bell Telephone Laboratories has shown that for carrier pulses the probability that noise of a given power will reduce signal pulses below half amplitude is less than the probability that this same noise will exceed half ampli- tude. This comes about from the fact that for effective cancellation there must be a 180° phase relationship between noise and pulse carrier. For this reason the slicing level should be set slightly above half amplitude for a carrier pulse system. The difference in performance between a perfect sheer and one with characteristics such as shown on Fig. 18 are as follows: For the perfect sheer no effects from noise or other disturbances are passed from one repeater to the next. For the case of the imperfect regenerator some ef- fects are passed on and so tend to accumulate in a chain of repeaters. To prevent this accumulated noise from building up to the breaking point of the system it is necessary to make the signal-to-noise ratio at each repeater somewhat better than that which would be required with the ideal sheer. For the case of random noise the required S/N ratio seems to be about 5 or 6 db above the theoretical value. This is due in part to sheer deficiency and in part to other system imperfections. 90 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 19o() CONCLUSIONS It is possible to build a simple device for regenerating pulses directly at microwave frequencies. A long chain of repeaters employing this regenerator should perform satisfactorily as long as the rms signal-to- noise ratio at each repeater is maintained at a value of 20 db or greater. There are a number of remaining problems which must be solved before we have a complete regenerative repeater. Some of these problems are: (1) Recovery of information for retiming from the incoming pulse train; (2) Automatic gain or level control to set the slicing level at each re- peater; (3) Simple, reliable, economical, broad-band microwave ampli- fiers. (4) Proper filters — both for transmitting and receiving. Traveling- wave tube development should eventually result in amplifiers which will meet all of the requirements set forth in (3) above. Any improve- ments which can be made in the regenerator without adding undue complications would also be advantageous. ACKNOWLEDGMENTS A. F. Dietrich assisted in setting up the equipment described here and in many other ways. The experiment would not have been possible with- out traveling-wave tubes and amplifiers which were obtained through the cooperation of M. E. Hines, C. C. Cutler and their associates. I wish to thank W. M. Goodall, and J. R. Pierce for many valuable suggestions. Crossbar Tandem as a Long Distance Switching System By A. O. ADAM (Manuscript received March 4, 1955) Major toll switching features are being added to the crossbar tandem switching system for use at many of the important long distance switching centers of the nationwide network. These include automatic selection of one of several alternate routes to a 'particular destination, storing and sending forward digits as required, highly flexible code conversion for transmitting digits different from those received, and a translating arrangement to select the most direct route to a destination. The system is designed to serve both operator and customer dialed long distance traffic. INTRODUCTION The crossbar tandem switching system,^ originally designed for switch- ing between local dial offices, will now play an important role in nation- wide dialing. New features are now available or are being developed that will permit this system to switch all types of traffic. As a result, crossbar [ tandem offices will have widespread use at many of the important switch- ing centers of the nationwide switching network. This paper briefly reviews the crossbar tandem switching system and its application for local switching, followed by discussion of the general aspects of the nationwide switching plan and of the major new features required to adapt crossbar tandem to this plan. CROSSBAR TANDEM OFFICES USED FOR LOCAL SWITCHING Crossbar tandem offices are now used in many of the large metropolitan areas throughout the country for interconnecting all types of local dial offices. In these applications they perform three major functions. Basi- cally, they permit economies in trunking by combining small amounts of 91 02 THE BELL SYSTEM TECHXIf AL JOURNAL, JANUARY 1956 traffic to and from the local offices into larger amounts for routing over common triuik groups to gain increased efficiency resulting in fewer over- all trunks. A second important function is to permit handling calls economically between different types of local offices which are not compatible from the standpoint of intercommunication by direct pulsing. Crossbar tandem offices serve to connect these offices and to supply the conversion from one type of pulsing to another where such incompatibilities exist. The third major function is that of centralization of equipment or services. For example, centralization of expensive charging equipment at a crossbar tandem office results in efficient use of such equipment and over-all lower cost as compared with furnishing this equipment at each local office requiring it. Examples of such equipment are remote control of zone registration and centralized automatic message accounting.^ Cen- tralization of other services such as weather bureau, time-of-day and similar services can be furnished. The first crossbar tandem offices were installed in 1941 in New York, Detroit and San Francisco. These offices were equipped to interconnect local panel and No. 1 crossbar central offices in the metropolitan areas, and to complete calls to manual central offices in the same areas. The war years slowed both development and production and it was not until the late 40's that many features now in use were placed in service. These later features enable customers in step-by-step local central offices on the fringes of the metropolitan areas to interconnect on a direct dialing basis with metropolitan area customers in panel, crossbar, manual and step- by-step central offices. This same development also permitted central offices in strictly step-by-step areas to be interconnected by a crossbar tandem office where direct interconnecting was not economical. Facilities were also made available in the crossbar tandem system for completing calls from switchboards where operators use dials or multifrequency key pulsing sets. Since a crossbar tandem office usually has access to all of the local offices in the area in which it is installed, it is attractive for handling short and long haul terminating traffic. The addition of toll terminal equipment at Gotham Tandem in New York City in 1947 permitted operators in New York State and northern New Jersey as well as distant operators to dial or key pulse directly into the tandem equipment for completion of calls to approximately 350 central offices in the New York metropolitan area. This method of completing these calls without the aid of the inward operators was a major advance in using tandem switch- ing ecjuipment for speeding completion of out-of-town calls. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 93 CROSSBAR TANDEM SWITCHING ARRANGEMENT The connections in a crossbar tandem office are established through crossbar switches mounted on incoming trunk link and outgoing office link frames shown on Fig. 1. The connections set up through these switches are controlled by equipment common to the crossbar tandem office which is held only long enough to set up each individual connec- tion. Senders and markers are the major common control circuits. The sender's function is to register the digits of the called number, transmit the called office code to the marker and then, as subsequently directed by the marker, control the outpulsing to the next office. The marker's function is to receive the code digits from the sender for translation, return information to the sender concerning the de- tails of the call, select an idle outgoing trunk to the called destination and close the transmission path through the crossbar switches from the incoming to the outgoing trunk. GENERAL ASPECTS OF NATIONWIDE DIALING Operator distance dialing, now used extensively throughout the country, as well as customer direct distance dialing are based on the division of the United States and Canada into numbering plan areas, interconnected by a national network through some 225 Control Switch- ing Points (CSP's) equipped with automatic toll switching systems. ^ An essential element of the nationwide dialing program is a universal numbering plan^ wherein each customer will have a distinctive number which does not conflict with the number of any other customer. The method employed is to divide the United States and Canada geographi- INCOMING TRUNK FROM ORIGINATING OFFICE TANDEM TRUNK TRUNK LINK FRAME 9 ? TRUNK LINK CONNECTOR SENDER LINK SENDER LINK CONTROL CIRCUIT SENDER OFFICE LINK FRAME <? 9 OFFICE LINK CONNECTOR J 4_ MARKER CONNECTOR OUTGOING TRUNK MARKER Fig. 1 — Crossbar tandem switching arrangement. 94 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 cally into more than 100 numbering plan areas and to give each of these a distinctive three digit code with either a 1 or as the middle digit. Each numbering plan area will contain 500 or fewer local central offices each of which will be assigned a distinctive three-digit office code. Thus each of the telephones in the United States and Canada will have, for distance dialing purposes, a distinct identity consisting of a three digit area code, an office code of two letters and a numeral, and a sta- tion number of four digits. Under this plan, a customer will dial 7 digits to reach another customer in the same numbering area and 10 digits to reach a customer in a different numbering area. A further reciuirement for nationwide dialing of long distance calls is a fundamental plan"* for automatic toll switching. The plan provides a systematic method of interconnecting all the local central offices and toll switching centers in the United States and Canada. As shown on Fig. 2, several local central offices or "end offices" are served by a single toll center or toll point that has trunks to a "home" primary center which serves a group of toll centers. Each primary center, has trunks to a "home" sectional center which serves a larger area of the country. Similuj-ly, the entire toll dialing territory is divided into eleven very large areas called regions, each having a regional center to serve all the sectional centers in the region. One of the regional centers, probably St. Louis, Missouri, will be designated the national center. The homing arrangements are such that it is not necessary for end offices, toll centers, toll points and primary centers to home on the next higher ranking office since the complete final route chain is not necessary. For example, end offices may be served directly from any of the higher ranking switch- ing centers also shown in Fig. 2. Collectively, the national center, the regional centers, the sectional centers and the primary centers will constitute the control switching points for nationwide dialing. The basic switching centers and homing arrangements are illustrated in Fig. 3. TANDEM CROSSBAR FEATURES FOR NATIONWIDE DIALING The broad objective in developing new features for crossbar tandem is to provide a toll switching system that can be used in cities where the large capacity and the full versatilit}^ of the No. 4 toll crossbar switching system-'' may not be economical. The application of crossbar tandem two-wire switching systems at primary and sectional centers has been made possible by the extended use of high speed carrier systems. The echoes at the 2-wire crossbar tandem switching offices can be effectively reduced by providing a high CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 95 office balance and by the use of impedance compensators and fixed pads. A well balanced two-wire switching system, proper assignment of inter- toll trunk losses, and the use of carrier circuits with high speed of propa- gation will permit through switching Mdth little or no impairment from an echo standpoint. The new features for crossbar tandem will provide arrangements necessary for operation at control switching points (CSP's). These in- clude automatic alternate routing, the ability to store and send forward TP e I I NC = NATIONAL CENTER RC = REGIONAL CENTER /\ SC = SECTIONAL CENTER ( J PC = PRIMARY CENTER Fig. 2 — Homing arrangement for local central offices and toll centers. TC = TOLL CENTER TP = TOLL POINT EG = END OFFICE 96 CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 97 digits as required, highly flexible code conversion (transmitting forward i different digits for the area or office code instead of the dialed digits), prefixing digits ahead of the called office code, and six-digit translation. ALTERNATE ROUTING The control switching points will be interconnected by a final or "backbone" network of intertoll trunks engineered so that very few calls will be delayed. In addition, direct circuits between individual switching offices of all classes will be provided as warranted by the traffic density. These are called "high-usage" groups and are not en- gineered to handle all the traffic offered to them during the busy hour. Traffic offered to a high-usage group which finds all trunks busy will be automatically rerouted to alternate routes®-^ consisting of other high- usage groups or to the final trunk group. The abi.ity of the crossbar tandem equipment at the control switching point to select one of several alternate routes automatically, when all choices in the first route are busy, contributes to the economy of the plant and provides additional protection against complete interruption of service when all circuits on a particular route are out of service. Fig. 4 shows a hypothetical example of alternate routing when a crossbar tandem office at South Bend, Indiana, receives a call destined for ^Youngstown, Ohio. To select an idle path, using this plan, the switching equipment at South Bend first tests the direct trunks to Youngstown. If these are all busy, it tests the direct trunks to Cleveland where the call would be completed over the final group to Youngstown. If the group to Cleveland is also busy, South Bend would test the group CHICAGO SOUTH BEND CROSSBAR TANDEM CLEVELAND -YOUNGSTOWN ITT5BURGH Fig. 4 — Toll network — alternate routing. 98 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 in UJ I- o o a. m t- < 2 a. m I- _j < mm •"^^ 1- I- 5m< 2cD2 _JOO ^^ < o O 1- gur-H I- < 2 ^ b UJ z o UJ Q rO UJ z i Ul UJ m X ll C\j I UJ (To: 1- X' Z o o o UJ <> m UJ (0 Hi' Hi' X oo 2-mmiiiii o (J X a UJ UJQ i£2 U.UJ °5 QO uicr -lU. _j <UJ UQ Ol Ul D CD Z ^ a. H (- o UJ Hi 2 in C 3 O o O +2 Hi ^ irt 2 O UJI- oo ZUJ aim I- to DO otr I I I I I I CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 99 to Pittsburgh and on its last attempt it would test the final group to Indianapolis. If the call were routed to Pittsburgh or Indianapolis, the switching equipment at these points would attempt by first choice and alternate routes to reach Youngstown. The final choice backbone route would be via Indianapolis, Chicago, St. Louis, Pittsburgh, Cleveland to Youngstown. Should all the trunks in any of the final groups tested be busy no further attempt to complete the call is made. It is unlikely that so many alternate routes would be provided in actual practice since crossbar tandem can test only a maximum of 240 trunks on each call and, in the case illustrated, the final trunk group to Indianapolis may be quite large. The method employed by the crossbar tandem marker in selecting the direct route and subsequent alternate routes is shown in simplified form on Fig. 5. As a result of the translating operation, the marker selects the first choice route relay, corresponding to the called destina- tion. Each route relay has a number of contacts which are connected to supply all the information recjuired for proper routing of the call. Several of these contacts are used to indicate the equipment location of the trunks and the number of trunks to be tested. The marker tests all of the trunks in the direct route and if they are busy, the search for an idle trunk continues in the first alternate route which is brought into play from the "route advance" cross-connection shown on the sketch. As many as three alternate routes in addition to the first choice route can be tested in this manner. STORING AND SENDING FORWARD DIGITS AS REQUIRED The crossbar tandem equipment at control switching points must store all the digits received and send forward as many as are required to complete the call. The called number recorded at a switching point is in the form of ABX-XXXX if the call is to be completed in the same numbering plan area. If the called destination is in another area, the area code XOX or XIX precedes the 7 digit number. The area codes XOX or XIX and the local office code ABX are the digits used for routing purposes and are sufficient to complete the call regardless of the number of switch- ing points involved. Each control switching point is arranged to ad- vance the call towards its destination when these codes are received. If the next switching point is not in the numbering area of the called telephone, the complete ten-digit number is needed to advance the call toward its destination. If the next switching point is in the num- 100 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 bering area of the called telephone the area code is not needed and seven digits will suffice for completing the call. For example, suppose a call is originated by a customer in South Bend, Indiana, destined for customer NAtional 4-1234 in Washington, D.C. If it is assumed that the route to Washington is via a switching center in Pittsburgh, then the crossbar tandem equipment at South Bend pulses forward to Pittsburgh 202-NA4-1234, 202 being the area code for the District of Columbia. Pittsburgh in turn will delete the area code and send NA4-1234 to the District of Columbia terminating area. As another example, suppose the crossbar tandem office at South Bend receives a call from some foreign area destined to a nearby step- by-step end office in Michigan. The crossbar tandem equipment re- ceives and stores a ten-digit number comprising the area code and the- seven digits for the office code and station number. Assuming that direct trunks to the step-by-step end office in Michigan are available, the area code and office code are deleted and the line number only is pulsed forward. To meet all conditions, the equipment is arranged to permit deletion of either the first three, four, five or six digits of a ten- digit number. CODE CONVERSION At the present time, some step-by-step primary centers reach other offices by the use of routing codes that are different from those assigned under the national numbering plan. This arrangement is used to obtain economies in switching equipment of the step-by-step plant and is accetpable with operator originated calls. However, with the intro- duction of customer direct distance dialing, it is essential that the codes used by customers be in accordance with the national numbering plan. The crossbar tandem control switching point must then automatically provide the routing codes needed by the intermediate step-by-step primary centers. This is accomplished by the code conversion feature which substitutes the arbitrary digits required to reach the called office through the step-by-step systems. Fig. 6 illustrates an application of this feature. It shows a crossbar tandem office arranged for completing calls through a step-by-step toll center to a local central office, GArden 8, in an adjacent area. A call reaching the crossbar tandem office for a customer in this office arrives with the national number, 218-GA8-1234. To complete this call, the crossbar tandem equipment deletes the area code 218 and pulses forward the local office code and number. If the « CROSSBAK TANDEM AS A TOLL SWITCHING SYSTEM 101 call is switched to an alternate route via the step-by-step primary center, it will be necessary for the crossbar tandem equipment to delete the area code 218 and substitute the arbitrary digits 062 to direct the call through the switches at the primary center, since the toll center requires the full seven digit number for completing the call. PREFIXING DIGITS It may be necessary to route a call from one area to another and back to the original area for completion. Such a situation arises on a call from Amarillo to Lubbock, Texas, both in area 915 when the crossbar tandem switching equipment finds all of the direct paths from Amarillo to Lubbock busy as illustrated on Fig. 7. The call could be routed to Lubbock via Oklahoma City which is in area 405. A seven-digit number for example, MAin 2-1234, is received in the crossbar tandem office at Amarillo. Assuming that the call is to be switched out of the 915 area through the 405 area and back to the 915 area for completion, it is necessary for the crossbar tandem office in Amarillo to prefix 915 to the MAin 2-1234 number so that the switching equipment in Oklahoma City will know that the call is for the 915 area and not for the 405 area. Prefixing digits may also be needed at crossbar tandem offices to route calls through step-by-step primary centers. The crossbar tandem office in Fig. 8 receives the seven digit number MA2-1234 for a call to a 701 AREA 218 AREA NUMBER RECEIVED 218-GA8-1234 CROSSBAR TANDEM NUMBER OUTPULSED 062-GA8-1234 STEP-BY-STEP PRIMARY CENTER ALTERNATE ROUTE DIRECT ROUTE GA8-I234 i^ GA8-t234 SX S TOLL CENTER GA8-1234 LOCAL CUSTOMER Fig. 6 — Code conversion. 102 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 customer in the Madison office in the same area. However, since the toll center needs the full seven digit number for completing the call and since the step-by-step switches at the primary center "use up" two digits (04) for its switching, the crossbar tandem equipment must prefix 04 to the seven digit number. METHOD OF DETERMINING DIGITS TO BE TRANSMITTED The circuitry involved for transmitting digits as received, prefixing, code conversion and for deletion involves both marker and sender functions. The senders have ten registers (1 to 10) for storing incoming digits and three registers (A A, AB, AC) for storing the arbitrary digits that are used for prefixing and code conversion. On a ten-digit call into a crossbar tandem switchmg center the area code XOX, the office code ABX and the station number XXXX are stored in the inpulsing or receiving registers of the sender. The code digits XOX-ABX are sent to the marker which translates them to determine which of the digits received by the sender should be outpulsed. It also determines whether arbitrary digits should be transmitted ahead of the digits received and, if so, the value of the arbitrary digits to be stored in the sender registers AA, AB and AC. Case 1 of Fig. 9 assumes that a ten-digit number has been stored in the sender registers 1 to 10 915 AREA INCOMING TOLL CALL LOCAL OFFICE AMARILLO CROSSBAR TANDEM OFFICE NUMBER RECEIVED MA2-1234 405 AREA ^< ■^ .-^ ^^o^^ .-^^^"J^^' OKLAHOMA CITY TOLL OFFICE LUBBOCK TOLL OFFICE MA 2 LOCAL CO. CUSTOMER MA 2-1234 Fig. 7 — Prefixing. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 103 and that the marker has mformed the sender the called number is to be sent as received. The outpulsing control circuit is connected to each register in turn through the steering circuit SI, S2, etc. and sends the digits stored. Case 2 illustrates a situation where the sender has stored ten digits in registers 1 to 10 and received information from the marker to delete the digits in registers 1 to 3 inclusive and to substitute the arbitrary digits stored in registers AA, AB and AC. The outpulsing circuit is first connected to register AA through steering circuit PSl, then to AB through PS2, continuing in a left to right sequence until all digits are outpulsed. Case 3 covers a condition where the sender has stored seven digits and has obtained information from the marker to prefix the two digits stored in registers AB and AC. Outpulsing begins at the AB register through steering circuit PS2 and then advances through steering circuit PS3 to the AC register, continuing in a left to right seciuence until all digits have been transmitted. These are only a few of the many combinations that are used to give the crossbar tandem control switching equipment complete pulsing flexibility. SIX-DIGIT TRANSLATION Six-digit translation will be another feature added to the crossbar tandem system. When only three digits are translated, it is necessary to direct all calls to a foreign area over a single route. The ability to trans- late six digits permits the establishment of two or more routes from the switching center to or towards the foreign area. This is shown in Fig. LOCAL OFFICE NUMBER OUTPULSED 04-MA2-1234 MADISON OFFICE MA2-I234 CROSSBAR TANDEM t ■ » ' ' — 1 n MA2- 1 1234 EIVED 4 4 1 1 TOLL CENTER — >- MADISON 2- 1234 STEP-BY-STEP PRIMARY CENTER Fig. 8 — Prefixing. 104 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 f/l 10 with Madison and Milwaukee, Wisconsin, in area 414 and Belle Plaine Crossbar Tandem in Chicago, Illinois, in area 312. An economical trunking plan may provide for direct circuits from Chicago to each place. If only three-digit translation were provided in the Chicago switching equipment, the route to both places would be selected as a result of the translation of the 414 area code alone and, therefore, calls to central offices reached through Madison, would need to be routed via Milwaukee. This involves not only the extra trunk mileage, ])ut also the use of an extra switching point. With six-digit translation, both the area code and the central office code are analyzed, making it possible to select the direct route to either city. Six-digit translation in crossbar tandem will involve primarily the use of a foreign area translator and a marker. The translator will have a capacity for translation of five foreign areas and for 60 routes to each area. Since the translator holding time is very short, one translator is sufficient to handle all of the calls requiring six-digit translation, but two are always provided for hazard and maintenance reasons. On a call requiring six-digit translation the first three digits are CASE 1 ^ DIGITS RECEIVED t 2 3 -IMPULSING 4 5 REGISTERS - 6 7 8 9 10 \ X X A B X X X X X . . ; i. OUTPULSING CONTROL ;Si : Sd S J Sa - S3 So . S / - So b» * oiu CASE 2 DIGITS RECEIVED OUTPULSING CONTROL DIGITS CODE CONVERTED AA AB AC ;: PS1 X' PS2 ;:PS3 ;:S4 ):S6 ~:S6 ;;S7 ::S8 10 59 ;:S10 CASE 3 DIGITS RECEIVED DIGITS PREFIXED AB AC B' C OUTPULSING CONTROL i PS2 : : P B PS3 •:SI ':S2 ::S3 :;S4 : : S5 :'S6 ■;S7 Fig. 9 — Method used for outpulsing digits. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 105 translated in the marker and the second three digits in a foreign area translator which is associated with the marker. Fig. 11 shows, in simpli- fied form, how this translation is accomplished. The first three digits, corresponding to the area code, are received by a relay code tree in the marker which translates it into one of a thousand code points. This code point is cross-connected to the particular relay of the five area relays A(3-A4 which has been assigned to the called area. A foreign area translator is now connected to the marker and a corre- sponding area relay is operated in it. The translator also receives the called office code from the sender via the marker and by means of a relay code tree similar to that in the marker translates the office code to one of a thousand code points. This code point plus the area relay is sufficient to determine the actual route to be used. As shown on the sketch, wires from each of the code points are threaded through trans- formers, two for each area. When the marker is ready to receive the route information, a surge of current is sent through one of these threaded wires which produces a voltage in the output winding to ionize the T- and U- tubes. Only the tubes associated with the area involved in the translation pass current to operate one each of the eight T- and U- relays. This information is passed to the marker and registered on corresponding tens and units relays. These operate a route relay which WISCONSIN MICH. J ILLINOIS CHICAGO = ' f BELLE \ 1 AREA IplaINeJ \312 I ^- — 1 I N D. ROUTE WITHOUT 6 DIGIT TRANSLATION ROUTE WITH 6 DIGIT TRANSLATION Fig. 10 — Six-digit translation. 106 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 11 — Method used for foreign area translation. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 107 provides all the information necessary for routing the call to the central office involved. CUSTOMER DIRECT DISTANCE DIALING Crossbar tandem will provide arrangements permitting customers in step-by-step offices to dial their own calls anywhere in the country. Centralized automatic message accounting previously mentioned will be used for charging purposes. While the basic plan for direct distance dialing provides for the dialing of either seven or ten digits, it will be necessary for the customer in step-by-step areas to prefix a three-digit directing code, such as 112, to the called number. This directing code is required to direct the call through the step-by-step switches to the crossbar tandem office so that the seven or ten digit number can be registered in the crossbar tandem office. When a customer in a step-by-step office originates a call to a distant customer whose national number is 915-CH3-1234, he first dials the directing code 112 and then the ten-digit number. The dialing of 112 causes the selectors in the step-by-step office to select an outgoing trunk to the crossbar tandem office. The incoming trunk in the crossbar tandem office has quick access to a three-digit register. The register must be connected during the interval between the last digit of the directing code and the first digit of the national number to insure registration of this number. This arrangement is used to permit the customer to dial all digits without delay and avoids the use of a second dial tone. If this arrangement were not used, the customer would be required to wait after dialing the 112 until the trunk in the tandem crossbar office could gain access to a sender through the sender link circuit which would then signal the customer to resume dialing by returning dial tone. After recording the 915 area code digits in the case assumed, the CH3-1234 portion of the number is registered directly in the tandem sender which has been connected to the trunk while the customer was dialing 915. When the sender is attached to the trunk, it signals the three-digit register to transfer the 915 area code digits to it via a con- nector circuit. Thus when dialing is complete, the entire number 915- CH3-1234 is registered in the sender. Crossbar tandem is being arranged to serve customers of panel and No. 1 crossbar offices for direct distance dialing. At the present time, ten digit direct distance dialing is not available to these customers because the digit storing equipments in these offices are limited to eight digits. Developments now under way, will provide arrangements for expanding the digit capacity in the local offices so that ultirnately 108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 calls from custoniers in panel and No. 1 crossbar offices may be routed through crossbar tandem cr other equivalent offices to telephones anywhere in the country. CONCLUSION The new features developed for crossbar tandem will adapt it to switching all types of traffic at many important switching centers of the nationwide toll network. Of the 225 important toll switching centers now contemplated, it is expected that about 80 of these will be ecjuipped with crossbar tandem. REFERENCES 1. Collis, R. E., Crossbar Tandem System, A.I.E.E. Trans., 69, pp. 997-1004, 1950. 2. King, G. v.. Centralized Automatic Message Accounting, B.S.T.J., 33, pp. 1331-1342, 1952. 3. Nunn, W. H., Nationwide Numbering Plan, B.S.T.J., 31, pp. 851-859, 1952. 4. Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, B. S.T.J. , 31, pp. 832-850, 1952. 5. Shipley, F. F., Automatic Toll Switching Systems, B.S.T.J., 31, pp. 860-882, 1952. 6. Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- ments in Alternate Routing Trunk Networks, B.S.T.J., 33, pp. 277-302, 1954. 7. Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Laboratories Record, 32, pp. 51-57, Feb. 1954. Growing Waves Due to Transverse Velocities By J. R. PIERCE and L. R. WALKER (Manuscript received March 30, 1955) This paper treats propagation of slow waves in two-dimensional neu- tralized electron floiv in which all electrons have the same velocity in the direction of propagation hut in which there are streams of two or more veloci- ties normal to the direction of propagation. In a finite beam in which ' electrons are reflected elastically at the boundaries and in which equal dc currents are carried by electrons with transverse velocities -\-Ui and — Wi , there is an antisi/mmetrical growing ivave if Up ~ {rUi/Wf and a symmetrical growing wave if y- i{Tu,/wy Here cop is plasma frequency for the total charge density and W is beam width. INTKODUCTION i It is well-known that there can be growing waves in electron flow when the flow is composed of several streams of electrons having different velocities in the direction of propagation of the waves. ' While Birdsall considers the case of growing waves in electron flow consisting of streams which cross one another, the growing waves which he finds apparently occur when two streams have different components of velocity in the direction of propagation. This paper shows that there can be growing waves in electron flow consisting of two or more streams with the same component of velocity in the direction of wave propagation but with different components of velocity transverse to the direction of propagation. Such growing Avaves can exist when the electric field varies in strength across the flow. Such waves could result in the amplification of noise fluctuations in electron ' flow. They could also be used to amplify signals. 109 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Actual electron flow as it occurs in practical tubes can exhibit trans- verse velocities. For instance, in Brillouin flow, ' • if we consider electron motion in a coordinate system rotating with the Larmor frequency we see that electrons with transverse velocities are free to cross the beam repeatedly, being reflected at the boundaries of the beam. The trans- verse \-elocities may be completely disorganized thermal velocities, or they may be larger and better-organized velocities due to aberrations at the edges of the cathode or at lenses or apertures. Two-dimensional Brillouin flow allows similar transverse motions. It would be difficult to treat the case of Brillouin or Brillouin-like flow with transverse velocities. Here, simpler cases with transverse velocities will be considered. The first case treated is that of infinite ion-neutra- lized two-dimensional flow with transverse velocities. The second case treated is that of two-dimensional flow in a beam of finite width in which the electrons are elastically reflected at the boundaries of the beam. Growing waves are found in both cases, and the rate of growth may be large. In the case of the finite beam both an antisymmetric mode and a symmetric mode are possible. Here, it appears, the current density required for a growing wave in the symmetric mode is about ^^ times as great as the current density required for a growing wa^•e in the anti- symmetric mode. Hence, as the current is increased, the first growing waves to arise might be antisymmetric modes, which could couple to a symmetrical resonator or helix only through a lack of symmetry or through high-level effects. 1 . Infinite two-dimensional flow Consider a two-dimensional problem in which the potential varies sinusoidally in the y direction, as exp{—j^z) in the z direction and as exp (jut) with time. Let there be two electron streams, each of a negative charge po and each moving with the velocity ?/o in the z direction, but with velocities Wi and —ih respectively in the y direction. Let us denote ac quantities pertaining to the first stream by subscripts 1 and ac quan- tities pertaining to the second stream by subscripts 2. The ac charge density will be denoted by p, the ac velocity in the y direction by y, and the ac velocity in the z direction by i. We will use linearized or small-signal equations of motion.^ We will denote differentiation with respect to ?/ by the operator D. The equation of continuity gives jupi = -D(piUi + po?yi) + j|8(piWo + pnii) (1.1)1 jcopo = -D{-p-iHi -\- pi)lj':d + il3(P2''o + Poi2) (1.2) t; GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 Let US define dx = i(co - ^u,) + u,D (1.3) do = ./(w - i8wo) - uj) (1.4) We can then rewrite (1.1) and (1.2) as f/iPi = Poi-Diji + j(3zi) (1.5) dopi = Pi^{ — Dy2 + .7/3i2) (1.0) We will assume that we are dealing ^^•ith slow waves and can use a po- tential V to describe the field. We can thus write the linearized equations of motion in the form r/iii = -j-^F (1.7) m d2h = -j-^V (1.8) m drlji = - DV (1.9) m d,y, = 1 DV (1.10) w From (1.5) to (1.10) we obtain ^m = ~ PoiD' - ^')V (1.11) m d'p2= --poiD'- ^')V (1.12) m Now, Poisson's equation is {D' - ^')V = _^L±£! (1.13) From (1.11) to (1.13) we obtain {D' - /3^)y = - Kco/ (^1 + ^^ (D' - /3^)7 (1.14) 9 ^ — Z— po 2 m Wp = e Here Wp is the plasma frequency for the charge of both beams. (1.15) 112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Either or else (2)' - /3')7 = — C0„" (c/l" + ^2") ^ 2 di^ d.} We will consider this second case. W(< should note from (1.3) and (1.4) that d{ = u^-D^ - (co - /5(/„)" + 2yD(co - |8?/.o)«i ^2^ = ?<i-D" - (co - ^ihf - 2jD{o^ - l3uo)ui di' + f/o' = 2{u{D' - (co - iSwo)'] rfiW = [uiD' + (co - /3;/„)T Thus, (1.17) becomes (1.16) (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) WD"" + (co - j8mo)2]^ If the quantities involved vary sinusoidally with y as cos ru or sin yy, -co, \u{lf - (co - /3ao)'] then Our equation becomes D' -7 (1.23) CO P L 1 + CO — jS'Uo T^Wi^ _ / co - 13^0 Y" \ 7^1 / (1.24) What happens if we have many transverse velocities? If we refer back to (1.14) we see that we will have an equation of the form 1 = E - 14 2^pn 2 I din + C?2n d^d ^ J ^^-^''^ "In (fin / Here cop„^ is a plasma frequency based on the density of electrons having transverse velocities ±Un . Equation (1.25) can be written (co - |(3//o)" "| i = E A^ 'M„2 r _ (g, - /3uo)2 -['^ L 7-'"n^ J (1.2()) GROWING WAVES DUE TO TRANSVERSE VELOCITIES 113 (u;-/3Uo Fig. 1 Suppose we plot the left-hand and the right-hand sides of (1.26) versus (co — ^Uo)- The general appearance of the left-hand and right-hand sides of (1.26) is indicated in Fig. 1 for the case of two velocities Un . There will always be two unattenuated waves at values of (w — /3wo) > y Ug where Ue is the extreme value of lu; these correspond to intersections 3 and 3' in Fig. 2. The other waves, two per value of Un , may be unat- tenuated or a pair of increasing and decreasing waves, depending on the values of the parameters. If CO pn -yhir? > 1 there will be at least one pair of increasing and decreasing waves. It is not clear what will happen for a Maxwellian distribution of veloci- ties. However, we must remember that various aberrations might give a very different, strongly peaked velocity distribution. Let us consider the amount of gain in the case of one pair of transverse velocities, ±i/i . The equation is now 2 2 7 Ui C0„2 [ 1 + CO — |3wo )•] [ ■ - (^OI (1.27) Let /5 = ^+i^ Wo Wo (1.28) 114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 1 .u 0.9 0.8 \ \ 0.7 0.6 \ \ \, \ ^ 0.5 0.4 0.3 \^ \ >s. \ V 0.2 \ > \ 0.1 \ \ \ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v2 m Fig. 2 This relation defines e. Equation (1.27) becomes 2 2 0}J 1 - e^ (1 + e^)^ ^'-''^ In Fig. 2, e is plotted versus the parameter y^Ui/oip^. We see that as the parameter falls below unity, e increases, at first rapidly, and then more slowly, reaching a value of ±1 as the parameter goes to zero (as cop' goes to infinity, for instance). It will be shown in Section 2 of this paper that these results for infinite flow are in some degree an approximation to the results for flow in narrow beams. It is therefore of interest to see what results they yield if applied to a beam of finite width. If the beam has a length L, the voltage gain is The gain G in db is G = 8.7 '^ € db Wo (1.30) (1.31) GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 Let the width of the beam be W. We let Thus, for n = 1, there is a half -cycle variation across the beam. From (1.31) and (1.32) G = 27.s(^^^\ne db (1.33) Now L/uo is the time it takes the electrons to go from one end of the beam to the other, while W/ui is the time it takes the electrons to cross the beam. If the electrons cross the beam A'' times iV = ^4 (1-34) Thus, G = 27.SNnedb (1.35) While for a given value of e the gain is higher if we make the phase vary many times across the beam, i.e., if we make n large, we should note that to get any gain at all we must have 2 . //iTTUlV 0)r> > (1.36) W If we increase oop , which is proportional to current density, so that cop passes through this value, the gain will rise sharply just after cOp" passes through this value and will rise less rapidly thereafter. .?. A Two-Dimensional Beam of Finite Width. Let us assume a beam of finite width in the ^/-direction ; the boundaries lying a,t y = ±^o • It will be assumed also that electrons incident upon these boundaries are elastically reflected, so that electrons of the incident stream (1 or 2) are converted into those of the other stream (2 or 1). The condition of elastic reflection implies that yi = -h (2.1) Zi = 22 Sit y = ±2/0 (2.2) and, in addition, that Pi = p2 at y = ±?/o . (2.3) since there is no change in the number of electrons at the boundary. 116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The equations of motion and of continuity (1.7-1.12) may be satisfied by introducing a single quantity, ^, such that V = dx dzV (2.4) ii = -J - /3 d, ^2^ (2.5) m zi = —j — di di\p (2.6) m yi=-d, d^Dyp (2.7) m 112=- di d^Di^ (2.8) m Pi m poiD' - ^') dirl^ (2.9) P2 = -- Po(i)' - n di'rl^ (2.10) m Then, if we introduce the symbol, 12, for co — jSuo yi + y^ = 2j-d,d2D^yp (2.11) ' m h- Z2 = 2j - di diUiD^ (2.12) m PI - P2 = 2j- po{D' - l3')uiQDi^ (2.13) m It is clear that if Drjy = D^xl^ = y = ±yo (2.14) the conditions for elastic reflection will be satisfied. The equation satis- fied by rf/ may now be found from Poisson's equation, (1-13), and is {D' - /3^) dx' di^P = '-^{D'- fi'){d,' + di)^l. we or {D' - ^')[{u,'D' + ny + coJiu.'D' - n')] = (2.15) which is of the sixth degree in D. So far four boundary conditions have, been imposed. The remaining necessary pair arise from matching the GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 internal fields to the external ones. For y > ijo V = Voe-'^'-e~^" (2.16) and Similarlv ^ + i37 = at 2/ = 2/0 dy dV — - ^V = at y = -7/0 (2.17) dy The most familiar procedure now would be to look for solutions of (2,15) of the form, e''^. This would give the sextic for c (c' - /3')[(WiV + nY + a;/(niV - n')] = (2.18) with the roots c = ±|8, ±ci , ±C2 , let us s^y. We could then express \p as a linear combination of these six solutions and adjust the coefficients to satisfy the six boundary equations. In this way a characteristic equa- tion for l3 would be obtained. From the S3anmetry of the problem this has the general form F(l3, Ci) = F(i3, C2), where Ci and Co are found from ; (2.18). The discussion of the problem in these terms is rather laborious and, if we are concerned mainly with examining qualitatively the onset of increasing waves, another approach serves better. From the symmetry of the equations and of the boundary conditions we see that there are solutions for \p (and consequently for V and p) which are even in y and again some which are odd in y. Consider first the even solutions. We will assume that there is an even function, ^i(y), periodic in y with period 2yo , which coincides with \l/(y) in the open interval, —yo<y<yo and that \pi(:y) has a Fourier cosine series repre- sentation : hiy) = E c„ cos \ny X„ = — n = 0, 1, 2, • • • (2.19) 1 yo yp inside the interval satisfies (2.15), so we assume that ypiiy) obeys (D^ - ^')[{u,'D' + ^'f + o.,\u,'D' - ^-)^, +00 (2.20) = Z) 5(2/ - 2m + lyo) where 6 is the familiar 5-function. Since D\p and D^\p are required to vanish at the ends of the interval and \l/, D'^ and Z)V are even it follows that all 118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 of these functions are continuous. We assume that xpi = \l/, D\pi = D\l/, DVi = D~\p, D% = D^yp and D% = D*xl/ at the ends of the intervals. From (2.20), Wi'D^i ^ -H as y ^ ijo . Since 2 8iy - 2m + lyo) = ^ + - £ (-1)" cos Ky (2.21) we obtain from (2.20) / 1 2?/oi/'i ,/32ff(i22 - Wp2) + 2i;(-l)" ^"^'"^ Since ^ + ^F = (Z) + /3)(t.x^Z)^ + fi^)V, using (2.4), the condition for matching to the external field, dV ^ + /37 = 0, dy yields, using D\p = DV = and Ui*D^\f/ = — i^, the relation (ui'D' + fi')Vi = 3^/3 at 2/ = 2/0 . Applying this to (2.22), we then obtain, finally, yo ^ 1 + 2Z r (^2 4- X„2)[(i22 - Ml2X„2)2 - cOp2(Q2 + ,,^2X„2)] (2.22) (2.23) For the odd solution we use a function, yp2(y), equal to ;/'(?/) in — //o < y < yo and representable by a sine series. To ensure the vanishing of D^p and 7)V at ?/ = ±?/o it is appropriate to use the functions, sin n„y, where Mn = (n -\- l'2)ir/yo . The period is now iyo and we define \p2(y) in /yo < y < 32/0 by the relation i;'2(2/) = ^{2yo — y) and in — 32/o < 2/ < — 2/o by ^2(2/) = ^{ — '^Uo — y)- Thus, we write 00 1^2(2/) = 2 C?n sin UnV Hn = (w + 3^)^7/0 ^2(2/) ^^i" ho supposed to satisfy GROWING "WAVES DUE TO TRANSVERSE VELOCITIES 119 +M (2.24) = 2 [^(y - 4m + lyo) - Ky - 4m - lyo)] m=— 00 The extended definition of i/'2 (outside — /yo <y < ijo) is such that we may again take \pi = \p, , D% = DV at the ends of the interval. ?/i*DVi is still equal to — }4 at ij = ijo . Now + 00 £ [5(y - 4m + iW - ^(y — 4m - l^/o)] (2.25) = — 2 (—1)" sin /i„?/ 2/0 so from (2.24) we may find v^L = -T (-l)"sin/xnj/ , ^ Matching to the external field as before gives and applied to (2.26) we have 00 /rfi 2 2\2 _y^ = y (^ - uinn) , . The equations (2.23) and (2.27) for the even and odd modes may be rewritten using the following reduced variables. . = ^« IT 1 _ Wj/0 _ Wo (2.23) becomes ^' 4- 2 y ^ (n' - k^ _ _ . and (2.27) transforms to „^ 2^ + (n + 3^)2 [{n + 1^)2 - /c2]2 - s\{n + 3^)^ + k'] (2 99) = — tt;? 120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We shall assume in considering (2.28) and (2.29) that the beam is sufficiently wide for the transit of an electron from one side to the other to take a few RF cycles. The number of cycles is in fact, coz/o/ttwi , and, hence, from the definition of z, we see that for values of A: less than 2, perhaps, z is certainly positive. Let us consider (2.29) first since it proves to be the simpler case. If we transfer the term ttz to the right hand side, it follo^^•s from the observa- tion that z is positive (for modest values of h), that it is necessary to make the sum negative. The sum may be studied qualitatively by sketch- ing in the k^ — d' plane the lines on which the individual terms go to infinity, given by [(n + 3^)^ - k'f 8' = (n -f K)' + k' (2.30) 3.5 Fig. 3 GROWING WAVES DUE TO TRANSVERSE VELOCITIES 121 77 0.4 0.3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (X/TT 1.6 1.8 2.0 2.2 2.4 Fig. 4 Fig. 3 shows a few such curves (n = 0, 1, 2). To the right of such curves the individual term in question is negative, except on the Hne, k^ = {n + V^) , where it attains the value of zero. Approaching the curves from the right the terms go to — oo . On the left of the curves the func- tion is positive and goes to + oo as the curve is approached from the 10 ... / / / / J, L / / / / / / L / / / / Y / /=, / A V / / / ' \ / A -0 1 / \ / y / \ ^^ A >< ■^ ^^ >C ^ "\ ^ '^ ^ 3 4 5 6 7 8 9 Fig. 5 J 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G left. Clearly in the regions marked + which lie to the left of every curve given by (2.30), the sum is positive and we cannot have roots. Let us examine the sum in the region to the right of the n = curve and to the left of all others. On the line, A;^ = J4» the sum is positive, since the first term is zero. On any other line, k' = constant, the sum goes from + °° at the n = 1 curve monotonically to — oo at the n = curve, so that somewhere it must pass through 0. This enables us to draw the zero- sum contours qualitatively in this region and they are indicated in Fig. 3. We are now in a position to follow the variation in the sum as k varies at fixed 5 . It is readily seen that for 5 < 0.25, because —wz is negative in the region under consideration, there will be four real roots, tw^o for positive, two for negative k. For 5' slightly greater than 0.25, the sum has Fig. 6A GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 a deep minimum for k = 0, so that there are still four real roots unless z is very large. For z fixed, as 5^ increases, the depth of the minimum de- creases and there will finally occur a 5" for which the minimum is so shal- low that two of the real roots disappear. Call z(0) the value of ziork = 0, write the sum as 2(5^ k^) and suppose that 2(5o^ 0) = —irziO), then for small k we have S(5^ e) = -«(0) + (6^ - 8o') §, + k'§,= -«(0) -"^ k do^ dk^ Ua as dB dk^ ^ = ^± / ".^(^-^0^) + '^ a/ dk' y The roots become complex when aA-2 S.2 J 2 (Ul/Uo) = do — 52 as d8^ dB Since Ui/uq may be considered small (say 10 per cent) it is sufficient to look for the values of 5o^. When k = we have -TZ = 2X) 2z z (n + y,y z^ + 52 irz" z'-\-in-\- y^r (n -1- y^y- - s' ' H ^ + i ^ \in + 3^)2 - 52 ^ (n + 1^)2 + zy (5 tan -Kb -\- z tanh irz) z" + 52 Fig. 4 shows the solution of this equation for various 2(0) or oiyo/iruo . Clearly the threshold 5 is rather insensitive to variations in uyo/ir^io . Equation (2.28) may be examined by a similar method, but here some complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; the n = term being of the form k^/k^ — 8^. The lowest critical region in 5^ is the neighborhood of the point fc^ = 6^ = ]^i, which is the intersec- tion of the n = and n = 1 lines. To obtain an idea of the behavior of 124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G the left hand side (l.h.s.) of (2.28) in this area we first see how the point k^ = f = 1^ can be approached so that the l.h.s. remains finite. If we put k^ = H + £ and a' = ^ + ce and expand the first two dominant terms of (2.28), then adjust c to keep the result finite as f -^ we find = 1 3^' - 5 ^ ~ 4 32^ + 1 c varies from — % to \i as z goes from to c» , changing sign at 2^ = %. Every curve for which the l.h.s. is constant makes quadratic contact with the Jine 5" — V3 = c(/v" — ]i) at Jc' = 5' = I/3. If we remember that the l.h.s. is positive for A;' = 0, < 5" < 1 and for A;^ = 1, < 5^ < 1, 1 2 lik 3 w-oX k^ / 1 y( I 3 SHADED AREAS // NEGATIVE yV X /' / // / /I / / / X \ n = i^v 3 3 Fig. 6B GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 since there are no negative terms in the sum for these ranges and again that the l.h.s. must change sign between the n = and n — I Unes for any k^ in the range < k^ < 1 (since it varies from T oo to ±0°), this information may be combined with that about the immediate vicinity of 5 = k = V^ to enable us to draw a Hue on which the l.h.s. is zero. This is indicated in Figs. 6A and 6B for small z and large z respec- tively. It will be seen that the zero curve and, in fact, all curves on which the l.h.s. is equal to a negative constant are required to have a vertical tangent at some point. This point may be above or below /c^ = ^ (de- pending upon the sign of c or the size of z) but always at a 3^ > ^. For 5 < H there are no regions where roots can arise as we can readily see by considering how the l.h.s. varies with k"^ at fixed 5^ For a fixed d^ > }/s we have, then, either for k^ > ]4 or k^ < V^, according to the size of z, a negative minimum which becomes indefinitely deep as 5^ -^ ^. Thus, since the negative terms on the right-hand side are not sensitive to small changes in 5^, we must expect to find, for a fixed value of the l.h.s., two real solutions of (2.28) for some values of 5^ and no real solutions for some larger value of 5 , since the negative minimum of the l.h.s. may be made as shallow as we like by increasing 6". By continuity then we expect to find pairs of complex roots in this region. Rather oddly these roots, which will exist certainly for 5' sufficiently close to V^ + 0, will disappear if 5^ is sufficiently increased. REFERENCES 1. L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, B. S. T. J., 28, pp. 23-51, Jan., 1949. 3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 4. G. G. Macfarlg,ne and H. G. Hay, Wave Propagation in a Slipping Stream of Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 5. P. Gurnard and H. Huber, Etude E.xp^rimentale de L'Interaction par Ondes de Chargd^d'Espace au Sein d'Un Faisceau Electronique se Deplagant dans Des Champs Electrique et Magn^tique Croisfe, Annales de Radio^lectricite, 7, pp. 252-278, Oct., 1952. 6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two Oblique Electron Streams, Technical Report No. 24, Electronics Research Laboratory, Stanford University. 7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van Nostrand, 1954. 9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950. Coupled Helices By J. S. COOK, R. KOMPFNER and C. F. QUATE (Received September 21, 1955) An analysis of coupled helices is presented, using the transmission line approach and also the field approach, with the objective of providing the tube designer and the microwave circuit engineer with a basis for approxi- mate calcidations. Devices based on the presence of only one mode of propa- gation are briefly described; and methods for establishing such a mode are given. Devices depending on the simultaneous presence of both modes, that is, depending on the beat wave phenomenon, are described; some experi- mental results are cited in support of the view that a novel and useful class of coupling elements has been discovered. CONTENTS 1. Introduction 129 2. Theory of Coupled Helices 132 2.1 Introduction 132 2.2 Transmission Line Equations 133 2.3 Solution for Synchronous Helices 135 2.4 Non-Synchronous Helix Solutions 137 2.5 A Look at the Fields 139 2.6 A Simple Estimate of b and x 141 2.7 Strength of Coupling versus Frequency 142 2.8 Field Solutions 144 . 2.9 Bifilar Helix 146 2.10 Effect of Dielectric Material between Helices 148 2.11 The Conditions for Maximum Power Transfer 151 2.12 Mode Impedance 152 3. Applications of Coupled Helices 154 3.1 Excitation of Pure Modes 156 3.1.1 Direct Excitation 156 3.1.2 Tapered Coupler 157 3.1.3 Stepped Coupler 158 3.2 Low Noise Transverse Field Amplifier 159 3.3 Dispersive Traveling Wave Tube 159 3.4 Devices Using Both Modes 161 3.4.1 Coupled Helix Transducer 161 3.4.2 Coupled-Helix Attenuator 165 4. Conclusion 167 Appendix I Solution of Field Equations 168 II Finding r I73 III Complete Power Transfer 175 127 128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 GLOSSARY OF SYMBOLS a Mean radius of inner helix h Mean radius of outer helix h Capacitive coupling coefficient Bio, 20 shunt susceptance of inner and outer helices, respectively Bi, 2 Shunt susceptance plus mutual susceptance of inner and outer helices, respectively, Bm + Bm , Boo + B^ Bm Mutual susceptance of two coupled helices c Velocity of light in free space d Radial separation between helices, h-a D Directivity of helix coupler E Electric field intensity F Maximum fraction of power transferable from one coupled helix to the other F(ya) Impedance parameter 7i, 2 RF current in inner and outer helix, respectively K Impedance in terms of longitudinal electric field on helix axis and axial power flow L ]\Iinimum axial distance required for maximum energy transfer from one coupled helix to the other, X6/2 Axial power flow along helix circuit Radial coordinate Radius where longitudinal component of electric field is zero for transverse mode (about midway between a and b) Return loss Radial separation betw^een helix and adjacent conducting shield Time RF potential of inner and outer helices, respectively • Inductive coupling coefficient Series reactance of inner and outer helices, respectively Series reactance plus mutual reactance of inner and outer helices, respectively, Xio + Xm , X20 + Xm Mutual reactance of two coupled helices Axial coordinate Impedance of inner and outer helix, respectively Attenuation constant of inner and outer helices, respectively General circuit phase constant; or mean circuit phase constant. Free space phase constant Axial phase constant of inner and outer helices in absence of coupling, V^ioXio , VBioXio p r f R s t F1.2 X Xva, 20 Xl, 2 Xm Z Zil, 2 Oil, 2 ^0 ^10. 20 COUPLED HELICES 129 181 , 2 May be considered as axial phase constant of inner and outer helices, respectively (Sft Beat phase constant jSc Coupling phase constant, (identical with ^b when /3i = JS2) I3ce Coupling phase constant when there is dielectric material be- tween the helices /3d Difference phase constant, [ /3i — /32 [ (8f Axial phase constant of single helix in presence of dielectric ^t, ( Axial phase constant of transverse and longitudinal modes, re- spectively 7 Radial phase constant jt, ( Radial phase constant of transverse and longitudinal modes, respectively r Axial propagation constant Tt. ( Axial propagation constant for transverse and longitudinal coupled-helix modes, respectively e Dielectric constant e' Relative dielectric constant, e/eq En Dielectric constant of free space X General circuit wavelength; or mean circuit wavelength, \/XiX2 Xo Free space wavelength Xi, 2 Axial wavelength on inner and outer helix, respectively X6 Beat wavelength Xc Coupling wavelength (identical with Xb when (5i = /So) yj/ Helix pitch angle i/'i, 2 Pitch angle of inner and outer helix, respectively CO Angular frequency 1. INTRODUCTION Since their first appearance, traveling-wave tubes have changed only very little. In particular, if we divide the tube, somewhat arbitrarily, into circuit and beam, the most widely used circuit is still the helix, and the most widely used transition from the circuits outside the tube to the circuit inside is from waveguide to a short stub or antenna which, in turn, is attached to the helix, either directly or through a few turns of increased pitch. Feedback of signal energy along the helix is prevented by means of loss, either distributed along the whole helix or localized somewhere near the middle. The helix is most often supported along its whole length by glass or ceramic rods, which also serve to carry a con- ducting coating ("aquadag"), acting as the localized loss. We therefore find the following circuit elements within the tube en- velope, fixed and inaccessible once and for all after it has been sealed off: 130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 1 . The helix itself, determining the beam voltage for optimum beam- circuit interaction ; 2. The helix ends and matching stubs, etc., all of which have to be positioned very precisely with relation to the waveguide circuits in order to obtain a reproducible match ; 3. The loss, in the form of "aquadag" on the support rods, which greatly influences the tube performance by its position and distril)ution. In spite of the enormous bandwidth over which the traveling-wave tube is potentially capable of operating — a feature new in the field of microwave amplifier tubes — it turns out that the positioning of the tube in the external circuits and the necessary matching adjustments are rather critical; moreover the overall bandwidths achieved are far short of the obtainable maximum. Another fact, experimentally observed and well-founded in theory, rounds off the situation: The electro-magnetic field surrounding a helix, i.e., the slow wave, under normal conditions, does not radiate, and is confined to the close vicinity of the helix, falling off in intensity nearly exponentially with distance from the helix. A typical traveling-wave tube, in which the helix is supported by ceramic rods, and the whole enclosed by the glass envelope, is thus practically inaccessible as far as RF fields are concerned, with the exception of the ends of the helix, where provision is made for matching to the outside circuits. Placing objects such as conductors, dielectrics or distributed loss close to the tube is, in general, observed to have no effect whatsoever. In the course of an experimental investigation into the propagation of space charge waves in electron beams it was desired to couple into a long helix at any point chosen along its length. Because of the feebleness of the RF fields outside the helix surrounded by the conventional sup- ports and the envelope, this seemed a rather difficult task. Nevertheless, if accomplished, such a coupling would have other and even more im- portant applications; and a good deal of thought was given to the problem. Coupled concentric helices were found to provide the solution to the problem of coupling into and out of a helix at any particular point, and to a number of other problems too. Concentric coupled helices have been considered by J. R. Pierce, who has ti'cated the problem mainly with transverse fields in mind. Such fields were thought to be useful in low-noise traveling-wave tube devices. Pierce's analysis treats the helices as transmission lines coupled uniformly over their length by means of nuitual distributed capacitance and inductance. Pierce also recognized that it is necessary to wind the COUPLED HELICES l,']! two helices in opposite directions in order to obtain well defined trans- verse and axial wave modes which are well separated in respect to their velocities of propagation. Pierce did not then give an estimate of the velocity separation which might be attainable with practical helices, nor did anybody (as far as we are aware) then know how strong a coupling one might obtain with such heUces. It was, therefore, a considerable (and gratifying) surprise^' ^ to find that concentric helices of practically realizable dimensions and separa- tions are, indeed, very strongly coupled when, and these are the im- portant points, (a) They have very nearly equal velocities of propagation when un- coupled, and when (b) They are wound in opposite senses. It was found that virtually complete power transfer from outer to inner helix (or vice versa) could be effected over a distance of the order of one helix wavelength (normally between i^fo and 3^^o of a free-space wavelength. It was also found that it was possible to make a transition from a co- axial transmission line to a short (outer) helix and thence through the glass surrounding an inner helix, which was fairly good over quite a con- siderable bandwidth. Such a transition also acted as a directional coupler, RF power coming from the coaxial line being transferred to the inner helix predominantly in one direction. Thus, one of the shortcomings of the "conventional" helix traveling- wave tube, namely the necessary built-in accuracy of the matching parameters, was overcome by means of the new type of coupler that might evolve around coupled helix-to-helix systems. Other constructional and functional possibilities appeared as the work progressed, such as coupled-helix attenuators, various tj^pes of broadband couplers, and schemes for exciting pure transverse (slow) or longitudinal (fast) waves on coupled helices. One central fact emerged from all these considerations: by placing part of the circuit outside the tube envelope with complete independence from the helix terminations inside the tube, coupled helices give back to the circuit designer a freedom comparable only with that obtained at much lower frequencies. For example, it now appears entirely possible to make one type of traveling wave tube to cover a variety of frequency bands, each band requiring merely different couplers or outside helices, the tube itself remaining unchanged. Moreover, one tube may now be made to fulfill a number of different 132 THE BELL SYSTEM TECHNICAI- JOURNAL, JANUARY 1956 functions; this is made possible by the freedom with which couplers and attenuators can be placed at any chosen point along the tube. Considerable work in this field has been done elsewhere. Reference will be made to it wherever possible. However, only that work with which the authors have been intimately connected will be fully reported here. In particular, the effect of the electron beam on the wave propaga- tion phenomena will not be considered. 2. THEORY OF COUPLED HELICES 2.1 Introduction In the past, considerable success has been attained in the under- standing of traveling wave tube behavior by means of the so-called "transmission-line" approach to the theory. In particular, J. R, Pierce used it in his initial analysis and was thus able to present the solution of the so-called traveling-wave tube equations in the form of 4 waves, one of which is an exponentially growing forward traveling wave basic to the operation of the tube as an amplifier. This transmission-line approach considers the helix — or any slow- wave circuit for that matter — as a transmission line with distributed capacitance and inductance with which an electron beam interacts. As the first approximation, the beam is assumed to be moving in an RF field of uniform intensity across the beam. In this way very simple expressions for the coupling parameter and gain, etc., are obtained, which give one a good appreciation of the physically relevant quantities. A number of factors, such as the effect of space charge, the non-uniform distribution of the electric field, the variation of circuit impedance with frequency, etc., can, in principle, be calculated and their effects can be superimposed, so to speak, on the relatively simple expressions deriving from the simple transmission line theory. This has, in fact, been done and is, from the design engineer's point of view, quite satisfactory. However, phj^sicists are bound to be unhappy over this state of affairs. In the beginning was Maxwell, and therefore the proper point to start from is Maxwell. So-called "Field" theories of traveling-w^ave tubes, based on Maxwell's equation, solved with the appropriate boundary conditions, have been worked out and their main importance is that they largely confirm the results obtained by the inexact transmission line theory. It is, however, in the nature of things that field theories cannot give answers in terms of COUPLED HELICES 133 simple closed expressions of any generality. The best that can be done is in the form of curves, with step-wise increases of particular param- eters. These can be of considerable value in particular cases, and when exactness is essential. In this paper we shall proceed by giving the "transmission-line" type theory first, together with the elaborations that are necessary to arrive at an estimate of the strength of coupling possible with coaxial helices. The "field" type theory will be used whenever the other theory fails, or is inadequate. Considerable physical insight can be gotten with the use of the transmission-line theory; nevertheless recourse to field theory is necessary in a number of cases, as will be seen. It will be noted that in all the calculations to be presented the presence of an electron beam is left out of account. This is done for two reasons: Its inclusion would enormously complicate the theory, and, as will eventually be shown, it would modify our conclusions only very slightly. Moreover, in practically all cases which we shall consider, the helices are so tightly coupled that the velocities of the two normal modes of propaga- tion are very different, as will be shown. Thus, only when the beam velocity is very near to either one or the other wave velocity, will growing-wave interaction take place between the beam and the helices. In this case conventional traveling wave tube theory may be used. A theory of coupled helices in the presence of an electron beam has been presented by Wade and Rynn,^ who treated the case of weakly coupled helices and arrived at conclusions not at variance with our views. 2.2 Transmission Line Equations Following Pierce we describe two lossless helices by their distributed series reactances Xio and A'20 and their distributed shunt susceptances Bio and ^20 . Thus their phase constants are /3io = V^ioA'io Let these helices be coupled by means of a mutual distributed reac- tance Xm and a mutual susceptance B^ , both of which are, in a way which will be described later, functions of the geometry. Let waves in the coupled system be described by the factor jut — Tj; e e \v here the F's are the propagation constants to be found. 134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The transmission line equations may be written: r/i - jB,V, + jB„y2 = rFi - iXi/i + jXJo = r/o - JB0V2 + jB„yi = TV2 - jXJa + jXJ, - where B, - 5io + 5« Bo = B20 + Bm X2 = X20 -f" Xm 1 1 and 1 2 are eliminated from the (2.2.1) and we find F2 ^ + (r- + XiBi + x^Bj Fi F2 (2.2.1) X\Bm + B%Xm + (r- + X2S2 + x^Bj XlBm + 5lX„ (2.2.2) (2.2.3) These two equations are then multipUed together and an expression for r of the 4th degree is obtained : r' + (XiBi + X2B2 + 2Z,„Bjr' + (X1Z2 - Xj){B,B2 - Bj) = We now define a number of dimensionless quantities: (2.2.4) B, BiB. Xm = h' = (eapacitive coupling coefficient)' = X = (inductive coupling coefficient) XiXo B\Xi = ^1, B2X2 = (82' X1B1X2B2 = 13^ = (mean phase constant) With these substitutions we obtain the general equation for T~ T' = 13' 2 \(3-r ^ I3{' ^ y 4v^2'^^/3i^ _ (2.2.5) + 26.r - (1 - .r-)(l - U') COUPLED HELICES 135 (2.2.6) If we make the same substitutions in (2.2.2) we find Fi T ZiL /3(/3i?> + /3o:r) . where the Z's are the impedances of the heUces, i.e., Z,. = VXJB, 2.3 Solution for Synchronous Helices Let us consider the particular case where (Si = (S-z = |S. From (2.2.5) we obtain r' = -I3\l + xb db (x + b)] (2.3.1) Each of the above values of T" characterizes a normal mode of propaga- tion involving both helices. The two square roots of each T" represent waves going in the positive and negative directions. We shall consider only the positive roots of T , denoted Tt and Tt , which represent the forward traveling waves. Ttj = i/3Vl + xb ± {x + b) (2.3.2) If a: > and 6 > I r, I > |/3i, I r,| < 1^1 Thus Vt represents a normal mode of propagation which is slower than the propagation velocity of either helix alone and can be called the "slow" wave. Similarly T( represents a "fast" wave. We shall find that, in fact, X and b are numerically equal in most cases of interest to us; we therefore write the expressions for the propagation constants r. = M^ + H(-^ + b)] (2.3.3) r. = Ml - Viix + b)] If we substitute (2.3.3) into (2.2.6) for the case where /3i = (82 = /3 and assume, for simplicity, that the helix self-impedances are equal, we find that for r = Tt Y% _ for r = T; F2 -— = -f 1 Yx ^ 136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Thus, the slow wave is characterized by equal voltages of unlike sign on the two helices, and the fast wave by equal voltages of like sign. It fol- lows that the electric field in the annular region between two such coupled concentric helices will be transverse for the slow wave and longitudinal for the fast. For this reason the slow and fast modes are often referred to as the transverse and longitudinal modes, respectively, as indi- cated by our subscripts. It should be noted here that we arbitrarily chose h and x positive. A different choice of signs cannot alter the fact that the transverse mode is the slower and the longitudinal mode is the faster of the two. Apart from the interest in the separate existence of the fast and slow waves as such, another object of interest is the phenomenon of the simul- taneous existence of both waves and the interference, or spatial beating, between them. Let V2 denote the voltage on the outer hehx; and let Vi , the voltage on the inner halix, be zero at z = 0. Then we have, omitting the common factor e'" , (2.3.4) Since at 2 = 0, Fi = 0, Vn = — V(^ . For the case we have considered we have found Fa = — V^ and Vn = V^ . We can write (2.3.4) as Fi = I {e~'^' - e-^n V, = ^ {e''^' + e-'n (2.3.5) F2 can be written = Ye-"'''''^''^''' cos [-jj^(r, - Vi)z\ In the case when x = 6, and /Si = /32 = /8 F2 = Ye"'^' cos Wiix + h)^z\ (2.3.6) Correspondingly, it can be shown that the voltage on the inner helix is y, = j\Tfr^^' sin Wiix + h)^z\ (2.3.7) The last tAvo equations exhibit clearly what we have called the spatial beat phenomonou, a wave-like transfer of power from one helix to thc^ COUPLED HELICES . 137 other and back. We started, arbitrarily, with all the voltage on the outer helix at 2 = 0, and none on the inner; after a distance, z', which makes the argument of the cosine x/2, there is no voltage on the outer helix and all is on the inner. To conform with published material let us define what we shall call the "coupling phase-constant" as ^, = ^{h + x) (2.3.8) From (2.3.3) we find that for (Si = ^2 = |S, and x = h, Tt - Ti = jl3c 2.4 Non-Synchronous Helix Solutions Let us now go back to the more general case where the propagation velocities of the (uncoupled) helices are not equal. Eciuation (2.2.5) can be written: Further, let us define (2.4.1) r- = -^- [1 + (1/2)A + xb ± V(l + xb)A + (1/4)A2 + (6 + xy] where L /3 _ In the case where x = h, (2.4.1) has an exact root. r,, , = j^ [Vl + A/4 ± 1/2 Va + (a; + by] (2.4.2) We shall be interested in the difference between Tt and Tt, Tt-Tf = j^ Va + (x + by- (2.4.3) Now we substitute for A and find Tt- Tc = j V(^i - ^2y + ^M& + 4' (2.4.4) Let us define the "beat phase-constant" as: Pb = V(/3i - /32)2 + nb + xy so that r, - r, = jA (2.4.5) (3a = \ i5i - iSo 138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 and call this the "difference phase-constant," i.e., the hase constant cor- responding to two uncoupled waves of the same frequency but differing phase velocities. We can thus state the relation between these phase constants : ^b' = &I + ^c (2.4.6) This relation is identical (except for notation) with expression (33) in S. E. Miller's paper. ^ In this paper Miller also gives expressions for the voltage amplitudes in two coupled transmission systems in the case of unequal phase velocities. It turns out that in such a case the power trans- fer from one system to the other is necessarily incomplete. This is of particular interest to us, in connection with a number of practical schemes. In our notation it is relatively simple, and we can state it by saying that the maximum fraction of power transferred is (2.4.7) or, in more detail, iS/ + iSc- (^1 - iS2)- + ^Kh + xY This relationship can be shown to be a good approximation from (2.2.6), (2.3.4), (2.4.2), on the assumption that h ^ x and Zx 'PH Z2 , and the further assumption that the system is lossless; that is, I 72 I ^ + I Fi I ^ = constant (2.4.8) We note that the phase velocity difference gives rise to two phenomena : It reduces the coupling w^avelength and it reduces the amount of power that can be transferred from one helix to the other. Something should be said about the case where the two helix imped- ances are not equal, since this, indeed, is usually the case with coupled concentric helices. Equation (2.4.8) becomes: I F2 1 _^ \Vx\_ ^ (3Qj^g^^j^^ (2.4.9) Z2 Z\ Using this relation it is found from (2.3.4) that F2 , /Zi FiT z, (1 ± Vl - /^) (2.4.10) When Ihis is combined with (2.2.6) it is found that the impedances droj) out with the voltages, and that "F" is a function of the |S's only. In other COUPLED HELICES 139 words, complete power transfer occurs when ,81 = /So regardless of the relative impedances of the helices. The reader will remember that (3io and (820 , not jSi and ^o , were defined as the phase constants of the helices in the absence of each other. If the assumption that h ^ x is maintained, it will be found that all of the de- rived relationships hold true when (Sno is substituted for /3„ . In other words, throughout the paper, /3i and /So may be treated as the phase con- stants of the inner and outer helices, respectively. In particular it should be noted that if these ciuantities are to be measured experimentally each helix must be kept in the same environment as if the helices were coupled ; onl}^ the other helix may be removed. That is, if there is dielectric in the annular region between the coupled helices, /Si and ^2 must each be measured in the presence of that dielectric. Miller also has treated the case of lossy coupled transmission systems. The expressions are lengthy and complicated and we believe that no substantial error is made in simply applying his conclusions to our case. If the attenuation constants ai and ao of the two transmission systems (helices) are equal, no change is required in our expressions; when they are unequal the total available power (in both helices) is most effectively reduced when ^4^'^l (2.4.11) Pc This fact may be made use of in designing coupled helix attenuators. 2.5 A Look at the Fields It may be advantageous to consider sketches of typical field distribu- tions in coupled helices, as in Fig. 2.1, before we go on to derive a quanti- tative estimate of the coupling factors actually obtainable in practice. Fig. 2.1(a) shows, diagrammatically, electric field lines when the coupled helices are excited in the fast or "longitudinal" mode. To set up this mode only, one has to supply voltages of like sign and equal ampli- tudes to both helices. For this reason, this mode is also sometimes called the "(+-f) mode." Fig. 2.1(b) shows the electric field lines when the helices are excited in the slow or "transverse" mode. This is the kind of field required in the transverse interaction type of traveling wave tube. In order to excite this mode it is necessary to supply voltages of equal amplitude and opposite signs to the helices and for this reason it is sometimes called the "(-| — ) mode." One way of exciting this mode consists in connecting one 140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 helix to one of the two conductors of a balanced transmission line ("Lecher"-line) and the other hehx to the other. Fig. 2.1(c) shows the electric field configuration when fast and slow modes are both present and equally strongly excited. We can imagine the two helices being excited by a voltage source connected to the outer (a) FAST WAVE (longitudinal) (b) SLOW WAVE (transverse) (C) fast and slow waves combined SHOWING SPATIAL "BEAT" PHENOMENON Fig. 2.1 — Typical electric field distributions in coupled coaxial helices when thej^ are excited in: (a) the in-phase or lonfritudinal mode, (b) the out-of-phase or transverse mode, and (c) both modes equally. COUPLED HELICES 141 helix only at the far left side of the sketch. One, perfectly legitimate, view of the situation is that the RF power, initially all on the outer helix, leaks into the inner helix because of the coupling between them, and then leaks back to the outer helix, and so forth. Apart from noting the appearance of the stationary spatial beat (or interference) phenomenon these additional facts are of interest: 1) It is a simple matter to excite such a beat- wave, for instance, by connecting a lead to either one or the other of the helices, and 2) It should be possible to discontinue either one of the helices, at points where there is no current (voltage) on it, without causing reflec- tions. 2.6 A Simple Estimate of h and x How strong a coupling can one expect from concentric helices in prac- tice? Quantitatively, this is expressed by the values of the coupling fac- tors X and h, which we shall now proceed to estimate. A first crude estimate is based on the fact that slow-wave fields are known to fall off in intensity somewhat as c where (3 is the phase con- stant of the wave and r the distance from the surface guiding the slow wave. Thus a unit charge placed, say, on the inner helix, will induce a charge of opposite sign and of magnitude -Pib-a) on the outer helix. Here h = mean radius of the outer helix and a = mean radius of the inner. We note that the shunt mutual admittance coupling factor is negative, irrespective of the directions in which the helices are wound. Because of the similarity of the magnetic and electric field distributions a current flowing on the inner helix will induce a simi- larly attenuated current, of amplitude on the outer helix. The direction of the induced current will depend on whether the helices are woimd in the same sense or not, and it turns out (as one can verify by reference to the low-freciuency case of coaxial coupled coils) that the series mutual impedance coupling factor is nega- tive when the helices are oppositely wound. In order to obtain the greatest possible coupling between concentric helices, both coupling factors should have the same sign. This then re- fiuires that the helices should be wound in opposite directions, as has been pointed out by Pierce. When the distance between the two helices goes to zero, that is to say, 142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 .if they lie in the same surface, it is clear that both coupling factors h and x will go to unity. As pointed out earlier in Section 2.3, the choice of sign for h is arbi- trary. However, once a sign for h has been chosen, the sign of x is neces- sarily the opposite when the helices are wound in the same direction, and vice versa. We shall choose, therefore, the sign of the latter depending on whether the helices are wound in the same direction or not. In the case of unequal velocities, (5, the propagation constant, would be given by 1^ = VM~2 (2.6.2) 2.7 Strength of Coupling versus Frequency The exponential variation of coupling factors with respect to frequency (since /3 = co/y) has an important consequence. Consider the expression for the coupling phase constant /3. = I3{b + x) (2.3.8) or l/3e| = 2/3^"^^'""^ (2.7.1) The coupling wavelength, which is defined as Ac is, therefore, 27r (2.7.2) or Xc- -e X, = ;^ g(2./x)u.-«) (2.7.3) where X is the (slowed-down) RF wavelength on either helix. It is con- venient to multiply both sides of (2.7.1) with a, the inner helix radius, in order to obtain a dimensionless relation between /3c and /3: ^,a = 2/3ac~^''"''°^"" (2.7.4) This relalion is j)l()Ued on Fig. 2.2 for several values of b/a. COUPLED HELICES 143 3.00 2.75 2.50 2.25 2.00 /3ca 1.75 1.50 1.25 i.OO 0.75 0.50 0.25 ^^ — - / / / / / / / / / l-y / / / / / J / / / /(/Jc3)max / / / / / 1 / / / / / ^ / b = 1.5 / / / / f ^ , V / ^^ '" \ 1 -\ "^^^ — 1 75 L 2.0 ■\ / "^ ■-^ 3.0 — - 0.5 1.5 2.0 2.5 /3a 3.0 3.5 4.0 4.5 5.0 Fig. 2.2 — Coupling pliase-constant plotted as a function of the single helix phase-constant for synchronous helices for several values of b/a. These curves are based on simple estimates made in Section 2.7. There are two opposing tendencies determining the actual physical length of a coupling beat-wavelength: 1) It tends to grow with the RF wavelength, being proportional to it in the first instance; 2) Because of the tighter coupling possible as the RF wavelength increases in relation to the heli.x-to-helix distance, the coupling beat- wavelength tends to shrink. Therefore, there is a region where these tendencies cancel each other, and where one would expect to find little change of the coupling beat- wavelength for a considerable change of RF freciuency. In other words, the "bandwidth" over which the beat-wavelength stays nearly constant can be large. This is a situation naturally very desirable and favorable for any device in which we rely on power transfer from one helix to the other by 144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 means of a length of overlap between them an integral number of half beat-wavelengths long. Ob^'iously, one will design the helices in such a way as to take advantage of this situation. Optimiun conditions are easily obtained by dijfferentiating ^c with respect to (3 and setting d^c/d^ equal to zero. This gives for the optimum conditions ^opt — 1 b — a (2.7.5) or Pc opt 2e h — a = 2e ')8opt (2.7.6) Equation (2.7.5), then, determines the ratio of the helix radii if it is re- quired that deviations from a chosen operating frequency shall have least effect. 2.8 Field Solutions In treating the problem of coaxial coupled helices from the transmis- sion line point of view one important fact has not been considered, namely, the dispersive character of the phase constants of the separate helices, /3i and fS-i . By dispersion we mean change of phase velocity with frequency. If the dispersion of the inner and outer helices were the same it would be of little consequence. It is well known, however, that the dispersion of a helical transmission line is a function of the ratio of helix radius to wavelength, and thus becomes a parameter to be considered. When the theory of wave propagation on a helix was solved by means of Maxwell's equations subject to the boundary condition of a helically conducting cylindrical sheath, the phenomenon of dispersion first made its appearance. It is clear, therefore, that a more complete theory of /i 'V^ 'TV Fig. 2.3 — ShoMtli liolix arrangement on which the field equations are based. COUPLED HELICES 145 coupled helices will require similar treatment, namely, Maxwell's equa- tions solved now with the boundary conditions of two cylindrical heli- cally conducting sheaths. As shown on Fig. 2.3, the inner helix is specified by its radius a and the angle 1^1 made by the direction of conductivity with a plane perpendicular to the axis; and the outer helix by its radius h (not to be confused with the mutual coupling coefficient 5) and its corresponding pitch angle i/'-j . We note here that oppositely wound helices require opposite signs for the angles \f/i and i/'o ; and, further, that helices with equal phase velocities will ha\'e pitch angles of about the same absolute magnitude. The method of solving Maxwell's equations subject to the above men- tioned boundary conditions is given in Appendix I. We restrict our- selves here to giving some of the results in graphical form. The most universally used parameter in traveling-wave tube design is a combination of parameters: /3oa cot \pi where (So = 27r/Xo , Xo being the free-space wavelength, a the radius of the inner helix, and xpi the pitch angle of the inner helix. The inner helix is chosen here in preference to the outer helix because, in practice, it will be part of a traveling-wave tube, that is to say, inside the tube envelope. Thus, it is not only less accessible and changeable, but determines the important aspects of a traveling-wave tube, such as gain, power output, and efficiency. The theory gives solutions in terms of radial propagation constants which we shall denote jt and yt (bj^ analogy with the transverse and longitudinal modes of the transmission line theory). These propagation constants are related to the axial propagation constants ^t and j3( by Of course, in transmission line theory there is no such thing as a radial propagation constant. The propagation constant derived there and de- noted r corresponds here to the axial propagation constant j^. By analogy with (2.4.5) the beat phase constant should be written How^ever, in practice ^0 is usually much smaller than j3 and Ave can there- fore write with little error iSfc = 7e — li for the beat phase constant. For practical purposes it is convenient to 146 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 J.OU _^^ 1 3i.Z0 COT ^2 _ „„„ ;:^ ^ COTV'i -0.90.. \ !^ -0.82,^ ^ :J^ 2.80 >^ <r Q 2.40 ^2.00 ^ /. ^ // / ^ |=,.25 1. 60 1.20 i /// 0.80 0.5 1.0 2.0 2.5 /io a COT ^, 3.0 3.5 4.0 4.5 Fig. 2.4.1 — Beat phase-constant plotted as a function of /3oa cot i^i • These curves result from the solution of the field equations given in the appendix. For hi a = 1.25. normalize in terms of the inner helix radius, a: jSbO 7<a — 7/a This has been plotted as a function of /5o a cot i/'i in Fig. 2.4, which should be compared with Fig. 2.2. It will be seen that there is considerable agreement between the results of the two methods, 2.9 Bifilar Helix The failure of the transmission line theory to take into account dis- persion is well illustrated in the case of the bifilar helix. Here we have two identical helices wound in the same sense, and at the same radius. If the two wires are fed in phase we have the normal mode characterized by the sheath helix model whose propagation constant is the familiar Curve A of Fig. 2.5. If the two wires of the helix are fed out of phase we have the bifilar mode; and, since that is a two wure transmission system, we shall have a TEM mode which, in the absence of dielectric, propa- gates along the wire with the velocity of light. Hence, the propagation constant for this mode is simplj' /3oa cot \p and gives rise to the horizontal COUPLED HELICES 147 1.80 1.60 (0 n 1.40 <5. I to t.OO 0.80 0.60 b. ^ ^>. "^ "a"'" A s ^ N. \. \^ i & \ \ ^ ■^ 0.82 w ^ = -0.98 COT^, ^ 0.90 ^V ^ // / \ \ v J, / \ \ \, t \ f \ f 0.5 1.0 1.5 2.0 2.5 /3oaCOTi^, 3.0 3.5 4.0 4.5 Fig. 2.4.2 — Beat phase-constant plotted as a function of /3oa cot ^i-i . These curves result from the solution of the field equations given in the appendix. For hia = 1.5. line of Curve B in Fig. 2.5. Again the coupling phase constant j3c is given by the difference of the individual phase constants: ^cO- — /3oa cot \f/ — ya (2.9.1) which is plotted in Fig. 2.6. Now note that when /So <3C 7 this equation is accurate, for it represents a solution of the field equations for the helix. From the simple unsophisticated transmission line point of view no coupling between the two helices would, of course, have been expected, since the two helices are identical in every way and their mutual capacity and inductance should then be equal and opposite. Experiments confirm the essential correctness of (2.9.1). In one experi- ment, which was performed to measure the coupling wavelength for the l)ifilar helices, we used helices with a cot 1/' = 3.49 and a radius of 0.036 cm which gave a value, at 3,000 mc, of ^oa cot i^ = 0.51 . In these experi- ments the coupling length, L, defined by (/3oa cot xp — 7a) — = TT a was measured to be 15.7o as compared to a value of 13.5a from Fig. 2.6. At 4,000 mc the measured coupling length was 14.6a as compared to 148 THE BELL vSYSTEM TECHNICAL JOURNAL, JANUARY 1956 1.20 b a 1.76 ^ ^^ ^ ^ X, / y ^ ^ S. X 1.00 / V \ ^. -\ / / \ P> •^.82 0.80 r — \ ^ <5. COT^ COT^ N ^ = -0.9 1 k^ "^^^ 0.90 (0 0.60 a >s^ X <0 '^ 0.40 ^ . 0.20 < D 5 1 1 5 2 .0 2 .5 ^1 3.0 3.5 4.0 4 Fig. 2.4.3 — Beat phase-constant plotted as a function of ^^a cot -^x . These curves result from the solution of the field equations given in the appendix. For hi a = 1.75. 12.6a computed from Fig. 2.6, thus confirming the theoretical prediction rather well. The slight increase in coupling length is attributable to the dielectric loading of the helices which were supported in quartz tubing. The dielectric tends to decrease the dispersion and hence reduce /3,. . This is discussed further in the next section. 2.10 Effect of Dielectric Material hetween Helices In many cases which are of interest in practice there is dielectric ma- terial between the helices. In particular when coupled helices are used with traveling-wave tubes, the tube envelope, which may be of glass, quartz, or ceramic, all but fills the space between the two helices. It is therefore of interest to know whether such dielectric makes any difference to the estimates at which we arrived earlier. We should not be surprised to find the coupling strengthened by the presence of the di- electric, because it is known that dielectrics tend to rob RF fields from the surrounding space, leading to an increase in the energy flow through the dielectric. On the other hand, tlio dielectric tends to bind the fields closer to the conducting medium. To find a qualitative answer to this question we have calculated the relative coupling phase constants for two sheath helices of infinite radius separated by a distance "d" for 1) COUPLED HELICES 149 1.00 b -a-^.u ^ ^^ ^ ^ j^ ^ COT Tp2 ^ ^ C 0.60 )^ 1 0.40 m y ^ COT }^, ^ 1 > -^ S^ ^ . , — -0. 90 =- -- V ^, i ^ ^0^ 98 0.20 1 1 ( 3 5 1 1 5 2 >oac 2.5 3 .0 3 .5 4 4. Fig. 2.4.4 — Beat phase-constant plotted as a function of /3oa cot ^i . These curves result from the solution of the field equations given in the appendix. For b/a = 2.0. the case with dielectric between them having a relative dielectric con- stant e' = 4, and 2) the case of no dielectric. The pitch angles of the two helices were \p and —xp, respectively; i.e., the helices were assumed to be synchronous, and wound in the opposite sense. ■ Fig. 2.7 shows a plot of the ratio of /3,,.//3, to ^d^ versus /3o (f//2) cotiA, 1.00 0.80 to n «5. 0.60 II m i 0.40 0.20 b a-o.u ^y ^ >< ^ y COT ^2 ^ \>^ ^-- COT 5^, -^ r /, ^ ^ ==^ "^^ N^ ^^ y^ ^ ^ ^ f/ ^ ^ N. ^ -c ).90 -^ r \ -^ _ -o.s ?8 , ^ 0.5 1.0 f.5 2.0 2.5 /JoacoT;^, 3.0 3.5 4.0 4.5 Fig. 2.4.5 — ■ Beat phase-constant plotted as a function of (3o« cot ^\ . These curves result from the solution of the field equations given in the appendix. For Va = 3.0. 150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 2.4 0.5 ).0 1.5 2.0 2. 5 3.0 3.5 /3oaCOT^ 4.0 4.5 Fig. 2.5 — Propagation constants for a bifilar helix plotted as a function of /3oa cot i/-! . The curves illustrate, (A) the dispersive character of the in-phase mode and, (B) the non-dispersive character of the out-of -phase mode. where ^^ is the coupling phase-constant in the presence of dielectric, /3j is the phase-constant of each helix alone in the presence of the same dielectric, ^c is the coupling phase-constant with no dielectric, and (3 is the phase constant of each helix in free space. In many cases of interest /3o(d/2) cot lA is greater than 1.2. Then 3£ + 1 " _2£' + 2_ g—(v'2« '+2-2)^0 (dl2) cot \l/ (2.10.1) Appearing in the same figure is a similar plot for the case when there is a conducting shield inside the inner helix and outside the outer, and separated a distance, "s," from the helices. Note that c? = 6 — a. It appears from these calculations that the effect of the presence of dielectric between the helices depends largely on the parameter /So (d/2) cot \{/. For values of this parameter larger than 0.3 the coupling wave- length tends to increase in terms of circuit wavelength. For values smaller than 0.3 the opposite tends to happen. Note that the curve representing (2.10.1) is a fair approximation down to /3o(c?/2) cot i/' = 0.6 to the curve representing the exact solution of the field equations. J. W. Sullivan, in unpublished work, has drawn similar conclusions. COUPLED HELICES 151 2.11 The Conditions for Maximum Power Transfer The transmission line theory has led us to expect that the most efficient power transfer will take place if the phase velocities on the two helices, prior to coupling, are the same. Again, this would be true were it not for the dispersion of the helices. To evaluate this effect we have used the field equation to determine the parameter of the coupled helices which gives maximum power transfer. To do this we searched for combinations of parameters which give an equal current flow in the helix sheath for either the longitudinal mode or the transverse mode. This was suggested by L. Stark, who reasoned that if the currents were equal for the indi- vidual modes the beat phenomenon would give points of zero RF current on the helix. The values of cot T/'2/cot 4/i which are required to produce this condi- tion are plotted in Fig. 2.8 for various values of b/a. Also there are shown values of cot ^2/cot \{/i required to give equal axial velocities for the helices before they are coupled. It can be seen that the uncoupled velocity of the inner helix must be slightly slower than that of the outer. A word of caution is* necessary for these curves have been plotted without considering the effects of dielectric loading, and this can have a rather marked effect on the parameters which we have been discussing. The significant point brought out by this calculation is that the optimum u.^o r N 0.24 0.20 / \ \ / N <D / \, / N / N^ ^ 0.12 \ \^ ~j ■^v CD ^■^^^ 0.08 f- ^- -^ 0.04 04 0.8 1.2 1.6 2.0 2.4 /3oaCOT J^, 2.8 3.2 3.6 4.0 Fig. 2.6 — The coupling phase-constant which results from the two possible modes of propagation on a bifilar helix shown as a function of jSoo cot i/-! . 152 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1956 2.6 2.4 2.2 2.0 i.8 1.6 u 1.4 i.2 1.0 0.8 0.6 0.4 0.2 PROPAGATION DIRECTION \ \ ^, \ L VA \ s PLANE SHEATH -^^^^"'^ XdiELECTRIC, HELICES \^^r e' CONDUCTING SHIELD \ \ \ s=oo \ \, APPROXIMATION ^^ ^, s \ "N '^ \ ^ "^^^ ■^^ o.t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 /iofcOT^^ Fig. 2.7 — ^ The effect of dielectric material between coupled infinite radius sheath helices on their relative coupling phase-constant shown as a function of fiod/2 cot \pi . The effect of shielding on this relation is also indicated. condition for coupling is not necessarily associated with equal \'elocities on the uncoupled helices. 2.12 Mode Impedance Before leaving the general theor_y of coupled helices something should be said regarding the impedance their modes present to an electron beam traveling either along their axis or through the annular space between them. The field solutions for cross woimd, coaxiall}^ coupled helices, which are given in Appendix I, have been used to compute the imped- ances of the transverse and longitudinal modes. The impedance, /v, is defined, as usual, in terms of the longitudinal field on the axis and the power flow along the system. COUPLED HELICES 153 K = F{ya) In Fig. 2.9, Fiya), for various I'atios of inner to outer radius, is plotted for both the transverse and longitudinal modes together with the value of F{ya) for the single helix {b/a = co). We see that the longitudinal mode has a higher impedance with cross wound coupled helices than does a single helix. We call attention here to the fact that this is the same phenomenon which is encountered in the contrawound helix^, where the structure consists of two oppositely wound helices of the same radius. As defined here, the transverse mode has a lower impedance than the single helix. This, however, is not the most significant comparison; for it is the transverse field midway between helices which is of interest in the transverse mode. The factor relating the impedance in terms of the transverse field between helices to the longitudinal field cni the axis is Er (f)/Ei(0), where f is the radius at which the longitudinal component of the electric field E^ , is zero for the transverse mode. This factor, plotted in Fig. 2.10 as a function of /3oa cot \l/r , shows that the impedance in. terms of the transverse field at f is interestingly high. 1.00 0.72 1.6 2.0 2.4 /3o a COT Ifi 4.0 Fig. 2.8 — The values of cot ^^./cot \pi required for complete power transfer plotted as a function of /3tia cot \pi for several values of b/a. For comparison, the value of cot ^2/cot \//i required for equal velocities on inner and outer helices is also shown. 154 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 F(ra) 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 • 3.0 2.5 2.0 1.5 1.0 0.5 0.5 (.0 (.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 /SoaCOTii', Fig. 2.9 — ■ Impedance parameter, F(ya), associated with both transverse and longitudinal modes shown for several values of b/a. Also shown is F{ya) for a single helix. It is also of interest to consider the impedance of the longitudinal mode in terms of the longitudinal field between the two helices. The factor, ^/(f)/£'/(0), relating this to the axial impedance is plotted in Fig. 2.11. We see that rather high impedances can also be obtained with the longitudinal field midway between helices. This, in conjunction with a hollow electron beam, should provide efficient amplification. LONGITUDINAL WAVE V COT U/2 \ \^. V \ \ \ "^ ^=-0.90 k \ \ COT U/^ V \ \ \ b.ooV ^ \ \ \ a \ \ ^ \ \ \ \-o\ \ \ \ \ \ \ \ > \ \ \ \ \ \ \ \ \ L \ i \ \ \ \ \ \ \ \ \ V \ \ ,25\ yp \ \ \ XT' \ \ \ \ \ ^ \ ' \ \ \ \ \ \ \ ' 1 \ \ \ \ \ \ \ \ k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ N, \ \ \ \ s > \ \ \ \, V, ^.0 \\ \ N s^5 \ \ 1 \ \ 1=1.2^ ^^^ ^ ^^ ^ '""'^-^ ^^ "^^ >., '- ^^ ^ ■ ^^ ^==^ 3. APPLICATION OF COUPLED HELICES When we come to describe devices which make use of coupled helices we find that they fall, quite naturally, into two separate classes. One COUPLED HELICES 155 class contains those devices which depend on the presence of only one of the two normal modes of propagation. The other class of devices depends on the simultaneous presence, in roughly equal amounts, of both normal modes of propagation, and is, in general, characterized by the words "spatial beating." Since spatial beating implies energy surging to and fro between inner and outer helix, there is no special problem in exciting both modes simultaneously. Power fed exclusively to one or the other /bo a COT jfi, Fiji;. 2.10 — The relation l)et\veen the impedance in terms of the transverse field between conpled helices excited in the out-of -phase mode, and the impedance in terms of the longitudinal field on the axis shown as a function of /3oa cot tpi . 156 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 helix will inevitably excite both modes equally. When it is desired to excite one mode exclusively a more difficult problem has to be solved. Therefore, in section 3.1 we shall first discuss methods of exciting one mode only before going on to discuss in sections 3.2 and 3.3 devices using one mode only. In section 3.4 we shall discuss devices depending on the simultaneous presence of both modes. 3.1 Excitation of Pure Modes 3.1.1 Direct Excitation In order to set up one or the other normal mode on coupled helices, voltages with specific phase and amplitudes (or corresponding currents) E|(f) E|(o) 10^ 5 10^ 10^ 10' 10 10" COT ip? ■ — = -0.90 COT i^, 1 / / 1 ' l-.o/ 1 L L 1 / J l\.2b / J / J / ^ '^ 3 A /ho a COT 1fi^ Fig. 2.11 — -The relation Ijetween the impedance in terms of the longitudinal field between couj)led helices excited in the in-phase mode, and the impedance in terms of the longitudinal field on the axis shown as a function of /3offl cot \pi . COUPLED HELICES 157 have to be supplied to each helix at the input end. A natural way of doing this might be by means of a two-conductor balanced transmission line (Lecher-line), one conductor being connected to the inner helix, the other to the outer helix. Such an arrangement would cause something like the transverse (-| — ) mode to be set up on the helices. If the two con- ductors and the balanced line can be shielded from each other starting some distance from the helices then it is, in principle, possible to intro- duce arbitrary amounts of extra delay into one of the conductors. A delay of one half period would then cause the longitudinal ( + + ) mode to be set up in the helices. Clearly such a coupling scheme would not be broad-band since a frequency-independent delay of one half period is not realizable. Other objections to both of these schemes are: Balanced lines are not generally used at microwave frequencies; it is difficult to bring leads through the envelope of a TWT without causing reflection of RF energy and without unduly encumbering the mechanical design of the tube plus circuits; both schemes are necessarily inexact because helices having different radii will, in general, require different voltages at either input in order to be excited in a pure mode. Thus the practicability, and success, of any general scheme based on the existence of a pure transverse or a pure longitudinal mode on coupled helices will depend to a large extent on whether elegant coupling means are available. Such means are indeed in existence as will be shown in the next sections. 3.1.2 Tapered Coupler A less direct but more elegant means of coupling an external circuit to either normal mode of a double helix arrangement is by the use of the so-called "tapered" coupler.^' ^' ^^ By appropriately tapering the relative propagation velocities of the inner and outer helices, outside the inter- action region, one can excite either normal mode by coupling to one helix only. The principle of this coupler is based on the fact that any two coupled transmission lines support two, and only two, normal modes, regardless of their relative phase velocities. These normal modes are characterized by unequal wave amplitudes on the two lines if the phase velocities are not equal. Indeed the greater the phase velocity difference and /or the smaller the coupling coefficient between the lines, the more their wave amplitudes diverge. Furthermore, the wave amplitude on the line with the slower phase velocity is greater for the out-of-phase or trans- verse normal mode, and the wave amplitude on the faster line is greater 158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G for the longitudinal normal mode. As the ratio of phase constant to coupling constant approaches infinity, the ratio of the wave amplitudes on the two lines does also. Finally, if the phase velocities of, or coupling between, two coupled helices are changed gradually along their length the normal modes existing on the pair roughly maintain their identity evan though they change their character. Thus, by properly tapering the phase velocities and coupling strength of any two coupled helices one can cause the two normal modes to become two separate waves, one existing on each helix. For instance, if one desires to extract a signal propagating in the in- phase, or longitudinal, normal mode from two concentric helices of equal phase velocity, one might gradually increase the pitch of the outer helix and decrease that of the inner, and at the same time increase the diameter of the outer helix to decrease the coupling, until the longitudinal mode exists as a wave on the outer helix only. At such a point the outer helix may be connected to a coaxial line and the signal brought out. This kind of coupler has the advantage of being frequency insensitive ; and, perhaps, operable over bandwidths upwards of two octaves. It has the disadvantage of being electrically, and sometimes physically, quite long. 3.1.3 Stepped Coupler There is yet a third way to excite only one normal mode on a double helix. This scheme consists of a short length at each end of the outer helix, for instance, which has a pitch slightly different from the rest. This has been called a "stepped" coupler. The principle of the stepped coupler is this: If two coupled transmis- sion lines have unlike phase velocities then a wave initiated in one line can never be completely transferred to the other, as has been shown in Section 2.4. The greater the velocity difference the less will be the maxi- mum transfer. One can choose a velocity difference such that the maxi- mum power transfer is just one half the initial power. It is a characteristic of incomplete power transfer that at the point where the maximum trans- fer occurs the waves on the two lines are exactly either in-phase or out-of- phase, depending on which helix was initially excited. Thus, the condi- tions for a normal mode on two equal-velocity helices can be produced at the maximum transfer point of two unlike velocity helices by initiating a wave on only one of them. If at that point the helix pitches are changed to give equal phase velocities in both helices, with equal current or volt- age amplitude on both helices, either one or the other of the two normal modes will be propagated on the two helices from there on. Although the COUPLED HELICES 159 pitch and length of such a stepped coupler are rather critical, the re- quirements are indicated in the equations in Section 2.4. The useful bandwidth of the stepped coupler is not as great as that of the tapered variety, but may be as much as an octave. It has however the advantage of being very much shorter and simpler than the tapered coupler. 3.2 Low-Noise Transverse-Field Amplifier r One application of coupled helices which has been suggested from the very beginning is for a transverse field amplifier with low noise factor. In such an amplifier the EF structure is required to produce a field which is purely transverse at the position of the beam. For the transverse mode there is always such a cylindrical surface where the longitudinal field is zero and this can be obtained from the field equation of Appendix II. In Fig. 3.1 we have plotted the value of the radius f at which the longi- tudinal field is zero for various parameters. The significant feature of this plot is that the radius which specifies zero longitudinal field is not constant with frequency. At frequencies away from the design frequency the electron beam will be in a position where interaction with longitudinal components might become important and thus shotnoise power will be introduced into the circuit. Thus the bandwidth of the amplifier over which it has a good noise factor would tend to be limited. However, this effect can be reduced by using the smallest practicable value of b/a. Section 2.12 indicates that the impedance of the transverse mode is very high, and thus this structure should be well suited for transverse field amplifiers. 3.3 Dispersive Traveling-Wave Tube Large bandwidth is not always essential in microwave amplifiers. In particular, the enormous bandwidth over which the traveling-wave tube is potentially capable of amplifying has so far found little application, while relatively narrow bandwidths (although quite wide by previous standards) are of immediate interest. Such a relatively narrow band, if it is an inherent electronic property of the tube, makes matching the tube to the external circuits easier. It may permit, for instance, the use of non-reciprocal attenuation by means of ferrites in the ferromagnetic resonance region. It obviates filters designed to deliberately reduce the band in certain applications. Last, but not least, it offers the possibility of trading bandwidth for gain and efficiency. A very simple method of making a traveling-wave tube narrow-band 160 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 0.5 1.W 1.8 ^ ^ 1.7 COT \p. _^ ^ ^ <^ T^ = -0. COT ^, 82^ ^ ^ ^ ^ l=- 1.6 ^ ^ ^ -0.90 ^ ^ ^ ^ * 1.5 — - -0.9 8 ' ^-' ' ^ ^ 1.4 -^ COT ^i'2 ^-^ = -0.82 COT UJ^ , -0.9j , ^- 1.3 . — "ZH ' 1 -0.98 1.2 _ "71 | = ,.25 — ^= CO' r 1//. — T-" H COT 5^, = -0.82 - -0.90 ■ ' / ^ 1 / / i.n -0 .98 ~ 1.0 1.5 2.0 2.5 3.0 /3o a COT j^. 3.5 4.0 4.5 5.0 Fig. 3.1 — The radius r at which the longitudinal field is zero for transversely excited coupled coaxial helices. is by using a dispersive circuit, (i.e. one in which the phase velocity varies significantly with frequency). Thus, we obtain an amplifier that can be limed by varying the beam voltage; being dispersive we should also expect a low group velocity and therefore higher circuit impedance. Calculations of the phase velocities of the normal modes of coupled concentric helices presented in the appendix show that the fast, longitu- dinal or (+ + ) mode is highly dispersive. Given the geometry of two such coupled helices and the relevant data on an electron beam, namely current, voltage and beam radius, it is possible to arrive at an estimate of the dependence of gain on frecjuency. Experiments with such a tube showed a Ijandwidth 3.8 times larger than the simple estimate would show. This we ascribe to the presence COUPLED HELICES 161 of the dielectric between the helices in the actual tube, and to the neglect of power propagated in the form of spatial harmonics. Nevertheless, the tube operated satisfactorily with distributed non- reciprocal ferrite attenuation along the whole helix and gave, at the center frequency of 4,500 mc/s more than 40 db stable gain. The gain fell to zero at 3,950 mc/s at one end of the band and at 4,980 mc/s at the other. The forward loss was 12 db. The backward loss was of the order of 50 db at the maximum gain frequency. 3.4 Devices Using Both Modes In this section we shall discuss applications of the coupled-helix princi- ple which depend for their function on the simultaneous presence of both the transverse and the longitudinal modes. When present in substantially equal magnitude a spatial beat-phenomenon takes place, that is, RF power transfers back and forth between inner and outer helix. Thus, there are points, periodic with distance along each helix, where there is substantially no current or voltage; at these points a helix can be terminated, cut-off, or connected to external circuits without detriment. The main object, then, of all devices discussed in this section is power transfer from one helix to the other; and, as will be seen, this can be ac- complished in a remarkably efficient, elegant, and broad-band manner. 3.4.1 Coupled-Helix Transducer It is, by now, a well known fact that a good match can be obtained between a coaxial line and a helix of proportions such as used in TWT's. A wire helix in free space has an effective impedance of the order of 100 ohms. A conducting shield near the helix, however, tends to reduce the helix impedance, and a value of 70 or even 50 ohms is easily attained. Pro\'ided that the transition region between the coaxial line and the helix does not present too abrupt a change in geometry or impedance, relatively good transitions, operable over bandwidths of several octaves, can l)e made, and are used in practice to feed into and out of tubes em- ploying helices such as TWT's and backward-wave oscillators. One particularly awkward point remains, namely, the necessity to lead the coaxial line through the tube envelope. This is a complication in manufacture and reciuires careful positioning and dimensioning of the helix and other tube parts. Coupled helices offer an opportunity to overcome this difficulty in the form of the so-called coupled-helix transducer, a sketch of which is shown in Fig. 3.2. As has been shown in Section 2.3, with helices having 162 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 the same velocity an overlap of one half of a beat wavelength will result in a 100 per cent power transfer from one helix to the other. A signal in- troduced into the outer helix at point A by means of the coaxial line will be all on the inner helix at point B, nothing remaining on the outer helix. At that point the outer helix can be discontinued, or cut off; since there is no power there, the seemingly violent discontinuity represented by the 'open" end of the helix will cause no reflection of power. In practice, un- fortunately, there are always imperfections to consider, and there will often be some power left at the end of the coupler helix. Thus, it is de- sirable to terminate the outer helix at this point non-reflectively, as, for instance, by a resistive element of the right value, or by connecting to it another matched coaxial line which in turn is then non-reflectively ter- minated. It will be seen, therefore, that the coupled-helix transducer can, in principle, be made into an efficient device for coupling RF energy from a coaxial line to a helix contained in a dielectric envelope such as a glass tube. The inner helix will be energized predominantly in one direction, namely, the one away from the input connection. Conversely, energy traveling initially in the inner helix will be transferred to the outer, and made available as output in the respective coaxial line. Such a coupled- helix transducer can be moved along the tube, if required. As long as the outer helix completely overlaps the inner, operation as described above should be assured. By this means a new flexibility in design, operation and adjustment of traveling-wave tubes is obtained which could not be achieved by any other known form of traveling-wave tube transducer. Naturally, the applications of the coupled-helix transducer are not restricted to TWT's only, nor to 100 per cent power transfer. To obtain Fig. 3.2 — A simple coupled helix transducer. COUPLED HELICES 1G3 power transfer of proportions other than 100 per cent two possibilities are open: either one can reduce the length of the synchronous coupling helix appropriately, or one can deliberately make the helices non-syn- chronous. In the latter case, a considerable measure of broad-banding can be obtained by making the length of overlap again equal to one half of a beat-wavelength, while the fraction of power transferred is deter- mined by the difference of the helix velocities according to 2.4.7. An application of the principle of the coupled-helix transducer to a variable delay line has been described by L. Stark in an unpublished memo- randum. Turning again to the complete power transfer case, we may ask: How broad is such a coupler? In Section 2.7 we have discussed how the radial falling-off of the RF energy near a helix can be used to broad-band coupled-helix devices which depend on relative constancy of beat-wavelength as frequency is varied. On the assumption that there exists a perfect broad-band match between a coaxial line and a helix, one can calculate the performance of a coupled-helix transducer of the type shown in Fig. 3.2. Let us define a center frequency co, at which the outer helix is exactly one half beat-wavelength, \b , long. If oj is the frequency of minimum beat wavelength then at frequencies coi and co2 , larger and smaller, respectively, than co, the outer helix will be a fraction 5 shorter than }i\b , (Section 2.7). Let a voltage amplitude, Vo , exist at the point where the outer helix is joined to the coaxial line. Then the magnitude of the voltage at the other end of the outer helix will be | F2 • sin (x5/2) | which means that the power has not been completely transferred to the inner helix. Let us assume complete reflection at this end of the outer helix. Then all but a fraction of the reflected power will be transferred to the inner helix in a reverse direction. Thus, we have a first estimate for the "directivity" defined as the ratio of forward to backward power (in db) introduced into the inner helix: D = 10 log sin" (3.4.1.1) We have assumed a perfect match between coaxial line and outer helix; thus the power reflected back into the coaxial line is proportional to sin^(x5/2). Thus the reflectivity defined as the ratio of reflected to incident power is given in db by i^ = 10 log sin' ^ (3.4.1.2) 164 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 For the sake of definiteness, let us choose actual figures: let /3a = 2.0. and hi a = 1.5. And let us, arbitrarily, demand that R always be less than -20 db. This gives sin (7r5/2) < 0.316 and 7r5/2 < 18.42° or 0.294 radians, 8 < 0.205. With the optimum value of (Sea = 1.47, this gives the mini- mum permissible value of I3ca of 1.47/(1 + 0.205) = 1.22. From the graph on Fig. 2.2 this corresponds to values of jSa of 1.00 and 3.50. Therefore, the reflected power is down 20 db over a frequency range of aj2/aji = 3,5 to one. Over the same range, the directivity is better than 10 to one. Suppose a directivity of better than 20 db were required. This requires sin (7r5/2) = 0.10, 8 = 0.0638 and is obtained over a fre- quency range of approximately two to one. Over the same range, the reflected power would be down by 40 db. In the above example the full bandwidth possibilities have not been used since the coupler has been assumed to have optimum length when jSctt is maximum. If the coupler is made longer so that when I3ca is maxi- mum it is electrically short of optimum to the extent permissible by the quality requirements, then the minimum allowable (S^a becomes even smaller. Thus, for h/a =1.5 and directivity 20 db or greater the rea- lizable bandwidth is nearly three to one. When the coupling helix is non-reflectively terminated at both ends, either by means of two coaxial lines or a coaxial line at one end and a resistive element at the other, the directivity is, ideally, infinite, irrespec- tive of frequency; and, similarly, there will be no reflections. The power transfer to the inner helix is simply proportional to cos (t8/2). Thus, under the conditions chosen for the example given above, the coupled- helix transducer can approach the ideal transducer over a considerable range of frequencies. So far, we have inspected the performance and bandwith of the coupled-helix transducer from the most optimistic theoretical point of view. Although a more realistic approach does not change the essence of our conclusions, it does modify them. For instance, we have neglected dispersion on the helices. Dispersion tends to reduce the maximum at- tainable bandwidth as can be seen if Fig. 2.4.2 rather than Fig. 2.2 is used in the example cited above. The dielectric that exists in the annular region between coupled concentric helices in most practical couplers may also affect the bandwidth. In practice, the performance^ of coupled-hc^lix transducers has been short of the ideal. In the first place, the match from a coaxial line to a helix is not perfect. Secondly, a not inappreciable fraction of the RF power on a real wire helix is propagated in the form of spatial harmonic COUPLED HELICES 165 28 26 24 22 20 18 )6 in _i LU m (4 u 12 10 r\ \ \ \ \ ' * / ' * / 1 t / [\ n [ 1 I 1 1 \j ^ \ Wf \ 1 \ \ I / I / \ / \1^ U~ / / / \ .' 1 A \J \- / \ \ A / Vi \ \ \ 1 1 / 1 p OUPLER DIRECTIVITY ETURN LOSS \ \ 1 J \ A V I / l 1.5 2.5 3 4 FREQUENCY IN KILOMEGACYCLES Fig. 3.3 — • The return loss and directivity of an experimental 100 per cent coupled-helix transducer. wave components which have variations with angle around the helix- axis, and coupling between such components on two helices wound in opposite directions must be small. Finally, there are the inevitable me- chanical inaccuracies and misalignments. Fig. 3.3 shows the results of measurements on a coupled-helix trans- ducer with no termination at the far end. 3.4.2 Coupled-Helix Attenuator In most TWT's the need arises for a region of heavy attenuation somewhere between input and output; this serves to isolate input and output, and prevents oscillations due to feedback along the circuit. Be- cause of the large bandwidth over which most TWT's are inherently capable of amplifying, substantial attenuation, say at least 60 db, is 166 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 required over a bandwidth of maybe 2 octaves, or even more. Further- more, such attenuation should present a very good match to a wave on the heHx, particularly to a wave traveling backwards from the output of the tube since such a wave will be amplified by the output section of the tube. Another requirement is that the attenuator should be physically as short as possible so as not to increase the length of the tube unneces- sarily. Finally, such attenuation might, with advantage, be made movable during the operation of the tube in order to obtain optimum performance, perhaps in respect of power output, or linearity, or some other aspect. Coupled-helix attenuators promise to perform these functions satis- factorily. A length of outer helix (synchronous with the inner helix) one half of a beat wavelength long, terminated at either end non-reflectively, forms a very simple, short, and elegant solution of the coupled-helix attenuator problem. A notable weakness of this form of attenuator is its relatively narrow bandwidth. Proceeding, as before, on the assumption that the attenuator is a fraction 8 larger or smaller than half a beat wavelength at frequencies coi and W2 on either side of the center frequency co, we find that the fraction of power transferred from the inner helix to the attenu- ator is then given by (1 — sin" (ir8/2)). The attenuation is thus simply A = sin^ (I) For helices of the same proportions as used before in Section 3.4.1, we find that this will give an attenuation of at least 20 db over a frequency band of two to one. At the center frequency, coo , the attenuation is in- finite; — in theory. Thus to get higher attenuation, it would be necessary to arrange for a sufficient number of such attenuators in tandem along the TWT. More- over, by properly staggering their lengths within certain ranges a wdder attenuation band may be achieved. The success of such a scheme largely depends on the ability to terminate the helix ends non-reflectively. Con- siderable work has been done in this direction, but complete success is not yet in sight. Another basically different scheme for a coupled-helix attenuator rests on the use of distributed attenuation along the coupling helix. The diffi- culty with any such scheme lies in the fact that unequal attenuation in the two coupled helices reduces the coupling between them and the moi'c they differ in respect to attenuation, the less the coupling. Naturally, one COUPLED HELICES 167 would wish to have as Httle attenuation as practicable associated with the inner helix (inside the TWT). This requires the attenuating element to be associated with the outer helix. Miller has shown that the maxi- mum total power reduction in coupled transmission systems is obtained when ai — 0:2 where ai and 012 are the attenuation constants in the respective systems, and ^b the beat phase constant. If the inner helix is assumed to be loss- less, the attenuation constant of the outer helix has to be effectively equal to the beat wave phase constant. It turns out that 60 db of attenuation requires about 3 beat wavelengths (in practice 10 to 20 helix wave- lengths). The total length of a typical TWT is only 3 or 4 times that, and it will be seen, therefore, that this scheme may not be practical as the only means of providing loss. Experiments carried out Avith outer helices of various resistivities and thicknesses by K. M. Poole (then at the Clarendon Laboratory, Oxford, England) tend to confirm this conclusion. P. D. Lacy" has described a coupled helix attenuator which uses a multifilar helix of resistance material together with a resistive sheath between the helices. Experiments were performed at Bell Telephone Laboratories with a TWT using a resistive sheath (graphite on paper) placed between the outer helix and the quartz tube enclosing the inner helix. The attenua- tions were found to be somewhat less than estimated theoretically. The attenuator helix was movable in the axial direction and it w^as instructive to observe the influence of attenuator position on the power output from the tube, particularly at the highest attainable power level. As one might expect, as the power level is raised, the attenuator has to be moA-ed nearer to the input end of the tube in order to obtain maximum gain and power output. In the limit, the attenuator helix has to be placed right close to the input end, a position which does not coincide with that for maximum low-level signal gain. Thus, the potential usefulness of the feature of mobility of coupled-helix elements has been demonstrated. 4. CONCLUSION In this paper we have made an attempt to develop and collect together a considerable body of information, partly in the form of equations, partl}^ in the form of graphs, which should be of some help to workers in the field of microwave tubes and devices. Because of the crudity of the assumptions, precise agreement between theory and experiment has not 168 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 been att-aiiu>(l iiur can it l)c expected. Nevertheless, the kind of physical phenomena occurring with coupled helices are, at least, qualitatively described here and should permit one to develop and construct various types of (lexices with fair chance of success. ACKNOWLEDGEMENTS As a final note the authors wish to express their appreciation for the patient work of Mrs. C. A. Lambert in computing the curves, and to G. E. Korb for taking the experimental data. Appendix i i. solution of field equations In this section there is presented the field equations for a transmission system consisting of two helices aligned with a common axis. The propa- gation properties and impedance of such a transmission system are dis- cussed for various ratios of the outer helix radius to the inner helix radius. This system is capable of propagating two modes and as previously pointed out one mode is characterized by a longitudinal field midway between the two helices and the other is characterized by a transverse field midway between the tw^o helices. The model which is to be treated and shown in Fig. 2.3 consists of an inner helix of radius a and pitch angle \pi which is coaxial with the outer helix of radius 6 and pitch angle \j/2 . The sheath helix model will be treated, wherein it is assumed that helices consist of infinitely thin sheaths which allow for ciuTent flow- only in the direction of the pitch angle \p. The components of the field in the region inside the inner helix, be- tween the two helices and outside the outer helix can be written as follows — inside the inner helix H,, = BrIoM (1) E., = B^hM (2) H,, = j - BMyr) (3) 7 Hr, = ^^ BMyr) (4) 7 E,, = -j "^ BMyr) (5) 7 Er, = -^ BJ,(rr) (()) 7 COUPLED HELICES 169 and between the two helices H,, = BMrr) + BJuirr) ' (7) E., = BJoiyr) + B^oiyr) (8) H,, = ^~ [B,h(yr) - B^^(yr)] (9) 7 Hr, = -^ [53/1(7/0 - BJuiyr)] (10) 7 E,, = - J ^ [B^hiyr) - BJuiyr)] (11) 7 Er, = -^ [BMyr) - BJv,{yr)\ (12) 7 and outside the outer hehx H.^ = B,Ko(yr) (13) E,, = 58/vo(7r) (14) ^.s = -J- BsK,{yr) (15) 7 Hr, = ^^ 5,Ki(7r) (16) 7 ^,, = i — BJuiyr) (17) 7 ^r« = ^^ 58Ki(7r) (18) 7 With the sheath helix model of current flow only in the direction of wires we can specify the usual boundary conditions that at the inner and outer helix radius the tangential electric field must be continuous and per- pendicular to the wires, whereas the tangential component of magnetic field parallel to the current flow must be continuous. These can be written as E, sin t/' + E^ cos ^ = (19) ' E, , E^ and (H, sin \f/ -f H^ cos \p) be equal on either side of the helix. By applying these conditions to the two helices the following equations are obtained for the various coefficients. 170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 First, we will define a more simple set of parameters. We will denote Io(ya) by /oi and h{yh) by /02 , etc. Further let us use the notation introduced by Humphrey, Kite and James" in his treatment of coaxial helices. Poi ^ laiKoi P02 = ToiKa2 Rq = I01K02 Pn = InKn P12 = InKu Ri = /iii^i2 and define a common factor (C.F.) by the equation r(/3oa cot hY p p (/3oa cot ^pif cot i/'z „ r, \_ (yay {jay cot t^i + Ro' — PoiP (20) .,] (21) With all of this we can now write for the coefficients of equations 1 through 18: y ju j8oa cot \pi 1 02 U iQoa cot 1^1 7oi/vi2 RiSoa cot i^i) y M ""to C.F. L ^4 _ _ • / £_ /3oa cot 1^1 /pi/ii r( B^~ -^ T M 7^ C.F. L" (7a)^ 5 5 (7a)'^ (/3oa cot 1^2)^ cot 1A2 p cot ;^i J P12 — jPo2 ■] B5 B, Bt Ro C.F. Ro — ((Soa cot xl/iY cot 1/' (7a^) (/3oa cot 1^2) cot l/' ;«'] (7a)^ 12 — -P02 B7 _ • . /£ i3oa cot lAi 1 /oi r 5; ~ "^ y M 7a C.F. K12 L Bs _ (|8oa cot i/'i)" cot 1/^2 /pi "" B2 {yay coT^i C.F.Po P02R1 — P02R1 - cot l/'2 cot i/'i cot l// cot \l/ 2R0 - P12R0 (22) (23) (24) (25) (26) (27) (28) The last equation necessary for the solution of our field problem is the transcendental equation for the propagation constant, 7, which can be COUPLED HELICES 171 written Ro — (i8o a cot \J/iY cot ^2 „ (yaY cot 4/1 [ = P02 - (jSo a cot \p2) D ? Vi -^ 12 Poi - (/3oa cot ^0" (yay _ (29) 11 The solutions of this equation are plotted in Fig. 4.1. There it is seen that there are two values of 7, one, yt , denoting the slow mode with transverse fields between helices and the other, yt , denoting the fast mode with longitudinal fields midway between the two helices. 5.0 4.S 4.0 3.5 3.0 ra 2.6 2.0 1.5 1.0 0.5 4 = 1.25 // / COT 5^2 COT ^1 0.82 0.90 0.98 ^ #■ /t // A na / f A y f / r / I / if < A / -<^^ ^ •y L -- •=**^ 0.5 1.0 1.6 2.0 2.5 3.0 /3o a COT yj 3.5 4.0 4.5 5.0 Fig. 4.1.1 —-The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of |3oa cot ^i-i for several values of hja = 1.25. 172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 These equations can now be used to compute the power flow as defined by P = }4 Re j E XH' which can be written in the form dA (30) r^;^(o)T L ^'p J fo © ^^-' ''' (31) where [F{ya, yb)] = (( W + (i8oa cot i/' {yar ^ /n^) (In' - /oi/2i)(C.F.)- - A'02' + 240 (C.F.)' (i8oa cot 1/^1)' (t«)'^ /or/n- r (80a cot ;^iY ' /Vl2" i^O - ya ((Soft cot i^i)' cot \p2 (ya)'' cot i/'i Rx - ) (/02/22 — /12') 4" (/ii — /01/21) , /p (/3oa cot 1A1)- cot \i/2 p Wp (^0^ ^'0^' "^2)'' p (ya)'^ cot i/'i (7a)^ ( - ) i'lInKu + /02/V22 + /22X02) — (2/iiKii + /01K21 + /21/voi) ot ^2)'^ p T (32) 2 , (^ofl cot l/'i) J ■> •'01 i- 7 r;; ^11 (l3oa cot - I (K02K22 — K12 ) — (K01K21 — Kn) .a, + (/3oa cot i^i)" A^ ■ 2 , (/Soa cot i/'2)" J 2 J. 2 (7a)'^ cot 1/^2 p J. I 02itl — -— r- i 12A0 cot 1^1 [/Vo2A'22 — /V12"] In (32) we find the power in the transverse mode by using values of COUPLED HELICES 173 5.0 0.5 2.0 2.5 3.0 /3o a COT y/ 5.0 Fig. 4.1.2 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of ^ofl cot \}/i when h/a — 1.50. yt obtained from (29) and similarly the power in the longittidinal mode is found by using values of yi . II. FINDING r When coaxial helices are used in a transverse field amplifier, only the transverse field mode is of interest and it is important that the helix parameters be adjusted such that there is no longitudinal field at some radius, f, where the cylindrical electron beam will be located. This condi- tion can be expressed by equating Ez to zero at r = f and from (8) BMyr) + B^,{yf) = (33) 174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 which can be written with (25) and (26) as (jSott cot ipiY cot \f/2 K(i2 Ri [ 02 ilO (7a)- cot \{/i = /oi Ri loM (/3oa cot \l/2)- ■I 02 — 7 rr, rn (34) Koiyf) This equation together with (29) enables one to evaluate f/a versus j8oa cot \l/i for various ratios of b/a and cot i^2/cot xpi . The results of these calculations are shown in Fig. 3.1. 5.0 4.5 4.0 3.5 3.0 7a 2.5 2,0 0.5 Fig. 4.1. .3 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a finiclion of 0oa cot \{/i when b/a = 1.75. i COUPLED HELICES 175 5.0 7a 2.0 2.5 3.0 /Oo <3 COT ^, 3.5 4.0 4.5 5.0 Fig. 4.1.4 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of /3oa cot yp\ when 6/a = 2.0. III. COMPLETE POWER TRANSFER For coupled heli.x applications we require the coupled helix parame- ters to be adjusted so that RF power fed into one helix alone will set up the transverse and longitudinal modes equal in amplitude. For this condition the power from the outer helix will transfer completely to the inner helix. The total current density can be written as the sum of the current in the longitudinal mode and the transverse mode. Thus for the inner helix we have -i&li J a = Jate-'''' + Jate .-J^<2 (35) 17G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 7?, 2.5 Fig. 4.1.5 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coa.xial sheath helices given as a function of /3oa cot i/-! when hi a = 3.0. and for the outer helix For complete power transfer we ask that •J hi — J hi when Jo is zero at the input {z = 0) or Jbt _ Jbt J at J at \ (36) \ (37) COUPLED HELICES 177 Now J at is equal to the discontinuity in the tangential component of magnetic field which can be written at r = a J at = {H,z cos ^i — //^5 sin \pi) — (H,i cos i/'i - H^o sin \f/i) \^'hich can be written as Ja( = - (H,i - H,3)a((cot i/'i + tau xj/i) slu \Pi (38) and similarily at r = h Jb( = — (H^7 — H,s)b({cot \p2 + tan 4^2) sin i/'2 (39) Equations (38) and (39) can be combined with (37) to give as the condi- tion for complete power transfer At = -At (40) where ^ = V (yay / ni) (T J^ _i- r V \( T? (/3oa cot <Ai)'^ cot 1^2 „ \ \ {yo,y cot i/'i / In (40) At is obtained by substituting jt into (41) and At is obtained by substituting 7 < into (41). The value of cot i/'o/cot i/'i necessary to satisfy (40) is plotted in Fig. 2.8. In addition to cot i/'o/cot i/'i it is necessary to determine the interference wavelength on the helices and this can be readily evaluated by consider- ing (36) which can now be written or /, = /,,.-«^'+^''-''^> cos ^ilJZ^ , (48) and J, = J.ce-'''^'^'^''"' cos M/3i^ (49) where we have defined iSfcO = {yta — jta) (50) This value of /S^ is plotted versus /3oa cot i/'i in Fig. 2.4. 178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 BIBLIOGRAPHY 1. J. R. Pierce, Traveling Wave Tubes, p. 44, Van Nostrand, 1950. 2. R. Kompfner, Experiments on Coupled Helices, A. E. R. E. Report No. G/M98, Sept., 1951. 3. R. Kompfner, Coupled Helices, paper presented at I. R. E. Electron Tube Conference, 1953, Stanford, Cal. 4. G. Wade and N. Rynn, Coupled Helices for Use in Traveling-Wave Tubes, I.R.E. Trans, on Electron Devices, Vol. ED-2, p. 15, July, 1955. 5. S. E. Miller, Coupled Wave Theory and Waveguide Applications, B.S.T.J., 33, pp. 677-693, 1954. 6. M. Chodorow and E. L. Chu, The Propagation Properties of Cross-Wound Twin Helices Suitable for Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Stanford Univ., June, 1953. 7. G. M. Branch, A New Slow Wave Structure for Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Stanford Univ., June, 1953. G. M. Branch, E.xperimental Observation of the Properties of Double Helix Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Univ. of Maine, June, 1954. 8. J. S. Cook, Tapered Velocity Couplers, B.S.T.J. 34, p. 807, 1955. 9. A. G. Fox, Wave Coupling by Warped Normal Modes, B.S.T.J., 34, p. 823, 1955. 10. W. H. Louisell, Analysis of the Single Tapered Mode Coupler, B.S.T.J., 34, p. 853. 11. B. L. Humphrey's, L. V. Kite, E. G. James, The Phase Velocity of Waves in a Double Helix, Report No. 9507, Research Lab. of G.E.C., England, Sept., 1948. 12. L. Stark, A Helical-Line Phase Shifter for Ultra-High Frequencies, Technical Report No. 59, Lincoln Laboratory, M.LT., Feb., 1954. 13. P. D. Lacy, Helix Coupled Traveling-Wave Tube, Electronics, 27, No. 11, Nov.. 1954. Statistical Techniques for Reducing the Experiment Time in Reliability Studies By MILTON SOBEL (Manuscript received September 19, 1955) Given two or more processes, the units from which fail in accordance with an exponential or delayed exponential law, the problem is to select the partic- ular process with the smallest failure rate. It is assumed that there is a com- mon guarantee period of zero or positive duration during which no failures occur. This guarantee period may be known or unknown. It is desired to accomplish the above goal in as short a time as possible without invalidating certain predetermined probability specifications. Three statistical techniques are considered for reducing the average experiment time needed to reach a decision. 1 . One technique is to increase the initial number of units put on test. This technique will substantially shorten the average experiment time. Its effect on the probability of a correct selection is generally negligible and in some cases there is no effect. 2. Another technique is to replace each failure immediately by a new unit from the same process. This replacement technique adds to the book- keeping of the test, but if any of the population variances is large (say in comparison with the guarantee period) then this technique will result in a substantial saving in the average experiment time. 3. A third technique is to use an appropriate sequential procedure. In many problems the sequential procedure results in a smaller average experi- ment time than the best non-sequential procedure regardless of the true failure rates. The amount of saving depends principally on the ^'distance'" between the smallest and second smallest failure rates. For the special case of two processes, tables are given to show the proba- bility of a correct selection and the average experiment time for each of three types of procedures. Numerical estimates of the relative efficiency of the procedures are given by computing the ratio of the average experiment time for two procedures of different type with the same initial sample size and satisfying the same probability specification. 179 180 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INTRODUCTION This paper is concerned with a study of the advantages and disad- vantages of three statistical techniques for reducing the average dura- tion of hfe tests. These techniques are: 1. Increasing the initial number of units on test. 2. Using a replacement technique. 3. Using a sequential procedure. To show the advantages of each of these techniques, we shall consider the problem of deciding which of two processes has the smaller failure rate. Three different types of procedures for making this decision will be considered. They are: Ri , A nonsequential, nonreplacement type of procedure E,2 , A nonsequential, replacement type of procedure Rs , A sequential, replacement type of procedure Within each type wq will consider different values of n, the initial number of units on test for each process. The effect of replacement is shown by comparing the average experiment time for procedures of type 1 and 2 with the same value of n and comparable probabilities of a correct selection. The effect of using a sequential rule is shown by com- paring the average experiment time for procedures of type 2 and 3 with the same value of n and comparable probabilities of a correct selection. ASSUMPTIONS 1. It is assumed that failure is clearly defined and that failures are recognized without any chance of error. 2. The lifetime of individual units from either population is assumed to follow an exponential density of the form f{x; e,g) =\ e-^^-")/" iov x -^ g f(x; e,g) = iorx<g where the location parameter g ^ represents the common guarantee period and the scale parameter 6 > represents the unknown parameter which distinguishes the two different processes. Let Ox ^ do denote the ordered values of the unknown parameter 6 for the two processes; then the ordered failure rates are given by Xi = 1/(01 + {/) ^ Xo = 1/(02 -f g) (2) 3. It is not known which process has the parameter di and which has the parameter dt . REDUCING TIME IN RELIABILITY STUDIES 181 4. The parameter g is assumed to be the same for both processes. It may be known or unknown. 5. The initial number n of units put on test is the same for both pro- cesses. 6. All units have independent lifetimes, i.e., the test environment is not such that the failure of one unit results in the failure of other units on test. 7. Replacements used in the test are assumed to come from the same population as the units they replace. If the replacement units have to sit on a shelf before being used then it is assumed that the replacements are not affected by shelf-aging. CONCLUSIONS 1. Increasing the initial sample size n has at most a negligible effect on the probability of a correct selection. It has a substantial effect on the average experiment time for all three types of procedures. If the value of n is doubled, then the average time is reduced to a value less than or equal to half of its original value. 2. The technique of replacement always reduces the average experi- ment time. This reduction is substantial when ^ = or when the popu- lation variance of either process is large compared to the value of g. This decrease in average experiment time must always be weighed against the disadvantage of an increase in bookkeeping and the necessity of having the replacement units available for use. 3. The sequential procedure enables the experimenter to make rational decisions as the evidence builds up without waiting for a predetermined number of failures. It has a shorter average experiment time than non- sequential procedures satisfying the same specification. This reduction brought about by the sequential procedure increases as the ratio a of the two failure rates increases. In addition the sequential procedure always terminates with a decision that is clfearly convincing on the basis of the observed results, i.e., the a posteriori probability of a correct selection is always large at the termination of the experiment. SPECIFICATION OF THE TEST Each of the three types of procedures is set up so as to satisfy the same specification described below. Let a denote the true value of the ratio 61/62 which by definition must be greater than, or equal to, one. It turns out that in each type of procedure the probability of a correct selection depends on 6i and 62 only through their ratio a. 182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 1. The experimenter is asked to specify the smallest value of a (say it is a* > I) that is worth detecting. Then the interval (1, a*) represents a zone of indifference such that if the true ratio a lies therein then we would still like to make a correct selection, but the loss due to a wrong selection in this case is negligible. 2. The experimenter is also asked to specify the minimum value P* > \'2 that he desires for the probability of a correct selection whenever a ^ a*. In each type of procedure the rules are set up so that the proba- bility of a correct selection for a = a* is as close to P* as possible without being less than P*. The two constants a* > 1 and \2 < P* < 1 are the only quantities specified by the experimenter. Together they make up the specification of the test procedure. EFFICIENCY If two procedures of different type have the same value of n and satisfy the same specification then we shall regard them as comparable and their relative efficiency will be measured by the ratio of their average experiment times. This ratio is a function of the true a but we shall consider it only for selected values of a, namely, a = 1, a = a* and a = CO . PROCEDURES OF TYPE Ri — • NONSEQUENTIAL, NONREPLACEMENT "The same number n of units are put on test for each of the two pro- cesses. Experimentation is continued until either one of the two samples produces a predetermined number r (r ^ n) of failures. Experimenta- tion is then stopped and the process with fewer than r failures is chosen to be the better one." Table I — Probability of a Correct Selection — Procedure Type Ri (a = 2, any g '^ 0, to be used to obtain r for a* = 2) n r = 1 r = 2 r = 3 r = i 1 0.667 . — . 2 0.667 0.733 — — 3 0.667 0.738 0.774 — 4 0.667 0.739 0.784 0.802 10 0.667 0.741 0.78!) 0.825 20 0.667 0.741 . 790 0.826 00 0.667 0.741 0.790 0.827 Note: The value for ?• = is obviously 0.500 for any n. REDUCING TIME IN RELIABILITY STUDIES 183 We shall assume that the number n of units put on test is determined by non -statistical considerations such as the availability of units, the availability of sockets, etc. Then the only unspecified number in the above procedure is the integer r. This can be determined from a table of probabilities of a correct selection to satisfy any given specification (a*, P*). If, for example, a* = 2 then we can enter Table I. If n is given to be 4 and we wish to meet the specification a* = 2, P* = 0.800 then we would enter Table I with n — 4 and select r = 4, it being the smallest value for which P ^ P*. The table above shows that for the given specification we would also have selected r = 4 for any value of n. In fact, we note that the proba- bility of a correct selection depends only slightly on n. The given value of n and the selected value of r then determine a particular procedure of type Ri , say, Ri(n, r). The average experiment time for each of several procedures R\{n, r) is given in Table II for the three critical values of the true ratio a, namely, a = \, a = a* and a = oo . Each of the entries has to be multi- plied by 6-1 , the smaller of the two d values, and added to the common guarantee period g. For n = oo the entry should be zero (-\-g) but it was found convenient to put in place of zero the leading term in the asymptotic expansion of the expectation in powers of I/71. Hence the entry for n = 00 can be used for any large n, say, n ^ 25 when r ^ 4. We note in Table II the undesirable feature that for each procedure the average experiment time increases with a for fixed 62 . For the se- quential procedure we shall see later that the average experiment time is greater at a = a* than at either a = 1 or a = 00 . This is intuitively more desirable since it means that the procedure spends more time when the choice is more difficult to make and less time when we are indifferent or when the choice is easy to make. PROCEDURES OF TYPE R2 — NONSEQUENTIAL, REPLACEMENT "Such procedures are carried out exactly as for procedures oiRi except that failures are immediately replaced by new units from the same population." To determine the appropriate value of r for the specification a* = 2, P* = 0.800 when g = we use the last row of Table I, i.e., the row marked n = ^ , and select r = 4. The probability of a correct selection for procedures of type Ro is exactly the same for all values of n and de- pends only on r. Furthermore, it agrees wdth the probability for pro- cedures of type Ri with n = co so that it is not necessary to prepare a separate table. PL, II H PM H K P o .^ Pi d 1 '^ I «3 'a 2 S CC Cl t-- o 00 r^ r-i o O '^ (N o ^ Ci o o »o CO o CO 00 ^ (M CD .-I o deo Oi r^ r^ CD ^ ^ lO o ■* CO i-i o rH 00(N CO CO CD CO o CO 00 CO »o o 00 o CO ^ o 1— I .— I o oco (M ^ t^ C5 (M t^ ■* O CO "tH lO O C^ T— I CO T-H O O O <M r^ »o r— I C2 CO 1—1 CO CO O CD (M t^ <M T-H o ^ O O O (M O CO CO 1— I CO o o CO 00 T-H o o lO 00 lO <M i— I O 1-1 o doo(M O lO-* (M ■* O o t^ t^ t^ 00 CO C^ CD ^ >— I O CD 1— I O O O O '-H t^ t^ CO iM -^ O t— I 1— t CD CO CD lO a; kO CO r-H o (M O O O O O T-H O O CO o o o o O O CO lO O lO o O lO CO (M T-H O O ^ o o o o o ^ t- CO (M t^ t^ CO t^ CD CO (M CD CD CO CD CD CO C^ >— I O O CD d> d> d CD d> d> d> s II a o o r^ vo o lO o O >0 CD (M lO (M O lO (M »-< 1-^ O O lO ooooooo --H iM CO "* O O 184 REDUCING TIME IN RELIABILITY STUDIES 185 Table III — Value of r Required to Meet the Specification (a*, P*) FOR Procedures of Type R2 (g = 0) a* p* 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 2.00 2.50 3.00 0.50 0.55 14 4 2 2 1 1 1 1 1 1 1 1 1 0.60 55 15 7 5 3 3 2 2 2 1 1 1 1 0.65 126 33 16 10 7 5 4 3 3 3 1 1 1 0.70 232 61 29 17 12 9 7 6 5 4 2 1 1 0.75 383 101 47 28 19 14 11 9 7 6 3 2 1 0.80 596 157 73 43 29 21 17 13 11 9 4 2 2 0.85 903 238 111 65 44 32 25 20 16 14 5 3 3 0.90 1381 363 169 100 67 49 37 30 25 21 8 5 4 0.95 2274 597 278 164 110 80 61 49 40 34 12 7 5 0.99 4549 1193 556 327 219 160 122 98 80 68 24 14 10 It i.s also unnecessary to prepare a separate table for the average ex- periment time for procedures of type R2 since for g = the exact values can be obtained by substituting the appropriate value of n in the ex- pressions appearing in Table II in the row marked n = oo . For example, for /( = 2, /• = 1 and a = 1 the exact value for ^ = is 0.500 62/2 = 0.250 62 , and for n = 3, r = 4, a = 00 the exact value for g = is 4.000 62/3 = 1.333 62 . It should be noted that for procedures of type R2 we need not restrict our attention to the cases r ^ n but can also con- sider r > //. Table III shows the value of r recjuired to meet the specilication (a*, F*) with a procedure of type R2 for various selected values of a* and P*. procedures of type R3 — sequential, replacement Let D{t) denote the absolute difference between the number of fail- ures produced by the two processes at any time t. The sequential pro- cedure is as follows: "Stop the test as soon as the inequality Dit) ^ In [P*/{1 - P*)] In a (3) is satisfied. Then select the population with the smaller number of fail- ures as the better one." To get the best results we will choose (a*, P*) so that the right hand member of the inequality (3) is an integer. Otherwise we would be operat- ing with a higher value of P* (or a smaller value of a*) than was specified. 186 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table IV — Average Experiment Time and Probability of a Correct Selection — Procedure Type R3 (a* = 2, P* = 0.800, ^ = 0) (Multiply each average time entry by d^) n a = 1 a = 2 a = 00 1 2.000 2.400 2.000 2 1.000 1.200 1.000 3 0.667 0.800 0.667 4 0.500 0.600 0.500 10 0.200 0.240 0.200 20 0.100 0.120 0.100 oc 2.000/w 2.400/n 2.000/n Probability 0.500 0.800 1.000 For example, we might choose a* = 2 and P* = 0.800. For procedures of type R3 the probability of a correct selection is again completely in- dependent of n; here it depends only on the true value of the ratio a. The average experiment time depends strongly on n and only to a limited extent on the true value of the ratio a. Table IV gives these quantities for a = 1, a = 2, and a = 00 for the particular specification a* = 2, p* = 0.800 and for the particular value ^ = 0. efficiency We are now in a position to compare the efficiency of two different types of procedures using the same value of n. The efficiency of Ri rela- tive to R2 is the reciprocal of the ratio of their average experiment time. This is given in Table V for a* = 2, P* = 0.800, r = 4 and n = 4, 10, 20 and 00 . By Table I the value P* = 0.800 is not attained for n < 4. In comparing the sequential and the nonsequential procedures it was found that the slight excesses in the last column of Table I over 0.800 Table V — Efficiency of Type Ri Relative to Type R2 {a* = 2, P* = 0.800, r = 4:,g = 0) { n a = 1 a = 2 a = 00 4 10 20 00 0.501 0.837 0.925 1.000 0.495 0.836 0.917 1.000 0.480 0.835 0.922 1.000 I REDUCING TIME IN RELIABILITY STUDIES 187 Table VI — Efficiency of («* = 2, P* Adjusted Ri Relative To R^ = 0.800, ^ = 0) n a = 1 a = 2 a = 00 4 10 20 00 0.615 0.754 0.818 0.873 0.575 0.708 0.768 0.822 0.419 0.528 0.573 0.612 had an effect on the efficiency. To make the procedures more comparable the values for r = 3 and r = 4 in Table I were averaged with values p and 1 — p computed so as to give a probability of exactly 0.800 at a = a*. The corresponding values for the average experiment time were then averaged with the same values p and 1 — p. The nonsequential pro- cedures so altered will be called "adjusted procedures." The efficiency of the adjusted Ri relative to Rz is given in Table VI. In Table VI the last row gives the efficiency of the adjusted procedure 7^2 relative to Rz . Thus we can separate out the advantage due to the replacement feature and the advantage due to the sequential fea- ture. Table VII gives these results in terms of percentage reduction of average experiment time. We note that the reduction due to the replacement feature alone is greatest for small n and essentially constant with a while the reduction Table VII — Per Cent Reduction in Average Experiment Time DUE TO Statistical Techniques (a* = 2,P* = 0.800, ^ = 0) a K Reduction due to Replacement Feature Alone Reduction due to Sequential Feature Alone Reduction due to both Replacement and Sequential Features 1 4 10 20 00 29.5 13.7 6.3 0.0 12.7 12.7 12.7 12.7 38.5 24.6 18.2 12.7 2 4 10 20 00 30.1 13.9 6.6 0.0 17.8 17.8 17.8 17.8 42.5 29.2 23.2 17.8 cc 4 10 20 00 31.5 13.6 6.3 0.0 38.8 38.8 38.8 38.8 58.1 47.2 42.7 38.8 188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 due to the sequential feature alone is greatest for large a and is inde- pendent of n. Hence if the initial sample size per process n is large we can disregard the replacement techniciue. On the other hand the true value of a is not known and hence the advantage of sequential experi- mentation should not be disregarded. The formulas used to compute the accompanying tables are given in Addendum 2. ACKNOWLEDGEMENT The author wishes to thank Miss Marilyn J. Huyett for considerable help in computing the tables in this paper. Thanks are also due to J. W. Tukey and other staff members for constructive criticism and numerical errors they have pointed out. Addendum 1 In this addendum we shall consider the more general problem of select- ing the best of k exponential populations treated on a higher mathemati- cal level. For k = 2 this reduces to the problem discussed above. DEFINITIONS AND ASSUMPTIONS There are given k populations H, (^ = 1, 2, • • • , k) such that the life- times of units taken from any of these populations are independent chance variables with the exponential density (1) with a common (known or unknown) location parameter g ^ 0. The distributions for the k popu- lations are identical except for the unknown scale parameter 6 > which may be different for the k different populations. We shall consider three different cases with regard to g. Case 1 : The parameter g has the value zero (g = 0). Case 2: The parameter g has a positive, known value (g > 0). Case 3: The parameter g is unknown (g ^ 0). Let the ordered values of the k scale parameters be denoted by di^ e.-^ ■■■ ^ dk (4) where equal values may be regarded as ordered in any arbitrary manner. At any time / each population has a certain number of failures associated with it. Let the ordered values of these integers be denoted by ri = ri{t) so that I ri g r2 ^ • • • ^ r-fc (5) ^ i REDUCING TIME IN RELIABILITY STUDIES 189 For each unit the life beyond its guarantee period will be referred to as its Poisson life. Let Li{t) denote the total amount of Poisson life observed up to time t in the population with Vi failures (z = 1, 2, • • • , fc). If two or more of the r^ are equal, say Vi = rj+i = • • • = r^+y , then we shall assign r, and L; to the population with the largest Poisson life, ri+i and L^+i to the population with the next largest, • • • , ri+_, and Lj+,- to the population with the smallest Poisson life. If there are two or more equal pairs (ri , Li) then these should be ordered by a random device giving equal probability to each ordering. Then the subscripts in (5) as well as those in (4) are in one-to-one correspondence with the k given populations. It should be noted that Li(t) ^ for all i and any time t ^ 0. The complete set of quantities Li{t) {i = 1, 2, • • • , k) need not be ordered. Let a = 61/62 so that, since the 6i are ordered, a ^ 1. We shall further assume that : 1 . The initial number n of units put on test is the same and the start- ing time is the same for each of the k populations. 2. Each replacement is assumed to be a new unit from the same popu- lation as the failure that it replaces. 3. Failures are assumed to be clearly recognizable without any chance of error. SPECIFICATIONS FOR CASE 1 : gf = Before experimentation starts the experimenter is asked to specify two constants a* and P* such that a* > 1 and l'^ < P* < 1. The procedure Ri = Rsin), which is defined in terms of the specified a* and P*, has the property that it will correctly select the population with the largest scale parameter with probability at least P* whenever a ^ a*. The initial number n of units put on test may either be fixed by nonstatistical con- siderations or may be determined by placing some restriction on the average experiment time function. Rule Rs : "Continue experimentation with replacement until the inequality k ^ ^*-(^.-a) ^ (1 _ p*)/p* (6) i=2 is satisfied. Then stop and select the population with the smallest num- ber of failures as the one having the largest scale parameter." 190 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Remarks 1. Since P* > Y2 then (1 — P*)/P* < 1 and hence no two popula- tions can have the same vahie ri at stopping time. 2. For A: = 2 the inequality (6) reduces to the inequalitj^ (3). 3. The procedure 7^3 terminates onl}^ at a failure time, never between failures, since the left member of (G) depends on t only through the quantities 7-i{t). 4. After experimentation is completed one can make, at the lOOP per cent confidence level, the confidence statement ds ^ di S a* 9, (or di/a"" ^ ds S e,) (7) where 6s is the scale parameter of the selected population. Numerical Illustrations »l/4 Suppose the preassigned constants are P* = 0.95 and a* = 19' 2.088 so that (1 - P*)/P* = ^9- Then for A; = 2 the procedure is to stop when r-i — ri ^ 4. For A; = 3 it is easy to check that the procedure reduces to the simple form: "Stop when ?'2 — ri ^ 5". For A; > 3 either calculations can be carried out as experimentation progresses or a table of stopping values can be constructed before experimentation starts. For A: = 4 and A; = 5 see Table VIII. In the above form the proposed rule is to stop Avhen, for at least one Table VIII — Sequential Rule for P* = 0.95, a* = 19 A: = 4 fc = 5 1/4 r2 — ri rs — ri n — ri 5 5 9 5 6 6 6 6 6 ri — ri ra — ri n — ci Ti — n 5 5 9 10 5 5 10 10 5 6 6 8 5 6 7 7 5 7 7 7 6 6 6 6 * Starred rows can be omitted without affecting the test since every integer in these rows is at least as great as the corresponding integer in the previous row. They are shown here to ilhistrate a systematic method which insures that all the necessary rows are included. REDUCING TIME IN RELIABILITY STUDIES 191 row (say row j) in the table, the observed row vector (r^ — Vi , Ts — Ti , ■ ■ ■ , Vk — z'l) is such that each comyonent is at least as large as the corresponding component of row j. Properties of Rs for k = 2 and g = For A- = 2 and ^ = the procedure Rs is an example of a Sequential Probability Ratio test as defined by A. Wald in his book.^ The Average Sample Number (ASN) function and the Operating Characteristics (OC) function for Rs can be obtained from the general formulae given by Wald. Both of these functions depend on di and 0-2 only through their ratio a. In our problem there is no excess over the boundary and hence Wald's approximation formulas are exact. When our problem is put into the Wald framework, the symmetry of our problem implies equal proba- bilities of type 1 and type 2 errors. The OC function takes on comple- mentary values for any point a = 61/62 and its reciprocal 62/61 . We shall therefore compute it only for a ^ 1 and denote it by P{a). For a > 1 the quantity P(a) denotes the probability of a correct selection for the true ratio a. The equation determining Wald's h function is 1 + a 1 + a for which the non-zero solution in h is easily computed to be h{a) = }^ (9) In a Hence we obtain from Wald's formula (3:43) in Reference 5 s a Pia) = -^^ (10) where s is the smallest integer greater than or equal to S = In [PV(1 - P*)]/ln a* (11) In particular, for a = 1"^, a* and 00 we have Pi^^) = 1/2, ^(«*) ^ P*, P(^) = 1 (12) ^\'e have written P(l"^) above for lim P{x) as x -^ 1 from the right. The procedure becomes more efficient if we choose P and a* so that *S' is an integer. Then s ^ S and P(a*) = P*. Letting F denote the total number of observed failures required to 192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 terminate the experiment we obtain for the ASN function and, in particular, for a = 1, oo E(F; 1) = s- and E{F; oo) = s (14) It is interesting to note that for s = 1 we obtain E{F; a) = 1 for all a ^ 1 (15) and that this result is exact since for s = 1 the right-hand member S \ of (3) is at most one and hence the procedure terminates with certainty ' immediately after the first failure. ' As a result of the exponential assumption, the assumption of replace- ; ment and the assumption that ^ = it follows that the intervals between \ failures are independently and identically distributed. For a single popu- ' lation the time interval between failures is an exponential chance vari- ; able. Hence, for two populations, the time interval is the minimum of j two exponentials which is again exponential. Letting r denote the i (chance) duration of a typical interval and letting T denote the (chance) j total time needed to terminate the procedure, Ave have E{T; a, 62) = E{F; a)E(r; a, d^) = E{F; a) (^^^ (f^) (16) I Hence Ave obtain from (13) and (14) E{T; a, 02) = - -^ ^^^ for a > 1 (17) n a — 1 a* + 1 E{T; 1, d,) = ^ and E{T; <^, 0,) = ^ (18> For the numerical illustration treated above Avith k = 2 we have na) = ^-^ (19) : P(l+) = ^; P(2.088) = 0.95; P(oo) = 1 (20) EiF-a) = 4^^4^ = 4 ^--+ Vy + '^ (21) a— la*-f-l a*-t-l E{F; 1) = 16.0; /iXF; 2.088) = 10.2; E{F; 00) = 4 (22), REDUCING TIME IN RELIABILITY STUDIES 193 E(T; 1, ^2) = — ; E{T; 2.088, 6^ = — ; n n (23) n For /.• > 2 the proposed procedure is an application of a general se- quential rule for selecting the best of A- populations which is treated in [1]. Proof that the probability specification is met and bounds on the probability of a correct decision can be found there. CASE 2: COMMON KNOWN ^ > In order to obtain the properties of the sec^uential procedure R:>. for this case it will be convenient to consider other sequential procedures. Let (S = 1/6-2 — 1/^1 so that, since the di are ordered, jS ^ 0. Let us assume that the experimenter can specify three constants a*, /3* and P* such that a* > 1, /3* > and ^ 2 < -P* < 1 ai^d a procedure is de- sired which will select the population with the largest scale parameter with probability at least P* whenever we have both a ^ a* and i3 ^ /3* The following procedure meets this specification. Rule Rs': "Continue experimentation with replacement until the inec^uality fi «*-(^i-'-i>e-^*(^i-^i)^ (l_p*)/p* (24) 1=2 is satisfied. Then stop and select the population with the smallest nimiber of failures as the one having the largest scale parameter. If, at stopping time, two or more populations have the same value ri then select that particular one of these with the largest Poisson life Li ." Remarks 1 . For k = 2 the inequality reduces to (r, - n) In a* + (Li - L2) 13* ^ In [P*/a - P*)] (25) If <7 = then Li = Li for all t and the procedure R/ reduces to R3 . 2. The procedure R/ may terminate not only at failures but also be- tween failures. 194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 3. The same inequality (24) can also be used if experimentation is carried on without replacement, one advantage of the latter being that there is less bookkeeping involved. In this case there is a possibility that the units will all fail before the inequality is satisfied so that the procedure is not yet completely defined for this case. One possibility in such a situation is to continue experimentation with new units from each population until the inequality is satisfied. Such a procedure will terminate in a finite time with probability one, i.e., Prob{ T > To} -^0 as To — > 00, and the probability specification will be satisfied. 4. A procedure R3 (ni , n-z , ■ • • , rik , ti , t2 , • • • , tk) using the same inequality (24) but based on dilTerent initial sample sizes and/or on different starting times for the initial samples also satisfies the above probability specification. In the case of different starting times it is required that the experimenter wait at least g units of time after the last initial sample is put on test before reaching any decision. 0. One disadvantage of R3 is that there is some (however remote) possibility of terminating while ri = r2 . This can be avoided by adding the condition r^ > n to (24) but, of course, the average experiment time is increased. Another way of avoiding this is to use the procedure R3 which depends only on the number of failures; the effect of using R3 when g > will be considered below. 6. The terms of the sum in (24) represent likelihood ratios. If at any time each term is less than unity then we shall regard the decision to select the population with n failures and Li units of Poisson life as opti- mal. Since (1 — P*)/P* < 1 then each term must be less than unity at termination. Properties of Procedure Rz for k = 2 p The OC and ASN functions for Rs will be approximated by comparing R3' with another procedure R/ defined below. We shall assume that P* is close to unity and that g is small enough (compared to d^) so that the probability of obtaining two failures within g imits of time is small enough to be negligible. Then we can write approximately at termination Li^nT - r,g {i = 1, 2, • • • , A:) (26) and Li - Li ^ (r, - r,)g (i = 2, 3, • • • , A:) (27) Substituting this in (24) and letting 5* = a* c^*" (28) suggests a new rule, say R/' , which we now define. REDUCING TIME IN RELIABILITY STUDIES 195 h'ule R/ "Continue experimentation with replacement until the inequality k X 6*-(^i-'-i) ^ (1 - P*)/P* (29) is satisfied. Then stop and select the population with n failures as the one with the largest scale parameter." For rule Rz" the experimenter need only specify P* and the smallest value 5* of the single parameter 8 = ^' e''''"''-''"''' = ae'^ (30) 62 that he desires to detect with probability at least P*. We shall approximate the OC and ASX function of R/' for k = 2 by computing them under the assumption that (27) holds at termina- tion. The results will be considered as an approximation for the OC and ASN functions respectively of R/ for /,■ = 2. The similarity of (29) and (6) immediately suggests that we might replace a* by 5* and a by 5 in the formulae for (6). To use the resulting expressions for R^ we would compute 5* as a function of a* and /3* by (28) and 5 as a function of a and /3 by (30). The similarity of (29) and (6) shows that Z„ (defined in Reference 5, page 170) under (27) with gr > is the same function of 5* and 5 as it is of a* and a when g = 0. To complete the justification of the above result it is sufficient to show that the individual increment ^ of Z„ is the same function of 5* and 8 under (27) with ^ > as it is of a* and a when ^ = 0. To keep the increments independent it is necessary to as- sociate each failure with the Poisson life that follows rather than with the Poisson life that precedes the failure. Neglecting the probability that any two failures occur ^^•ithin g units of time we have two values for z, namely ^ -(.nt-g)/ei -ntl$2 z = log^^^ = -log 5 (31) and, interchanging 61 and ^2 , gives z — log 5. Moreover 196 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 r r - e-(— «)/«v"^^^^ dx dy Jg Jg 6-1 Prob \z = -logSj ^2 -0[92(n-l)+9l"l/9lfl2 _i_ ^1 -H9in+Bi(n-l)]l9ie2,) - e + - e /o9\ 1 + 5 Thus the OC and ASN functions under (27) with g > bear the same relation to 5* and 5 as they do to a* and a when ^ = 0. Hence, letting w denote the smallest integer greater than or equal to ^ In [P*/(l - P*)] ^ \n[P*/{l-P*)] In 8* gl3* + In a* ^' ' we can write (omitting P* in the rule description) | 7^15; /?/ («*, /5*){ ^ P{5; /^.^"(S*)! ^ ^-^^^ (34) <w. I ■ — - tor 5 > 1 {So) ^ \8 - l/\5"' +1/ w~ for 5=1 W'e can approximate the average time between failures by I and the average experiment time by « E{T; /?/(«*, ^*)} ^ E{F; R,'(a*, 0*)\ [^.^ f^'^ _^ ^'^, (37) n{Oi -T 02 -f- zg) Since 5 ^ 1 then 5"(1 + 5") is an increasing function of w and by (33) it is a non-increashig function of 5*. By (28) 5* ^ a* and hence, if we disregard the approximation (34), P{8; AV(«*)1 - ^!^{py^/_p.^y..n ^* ^ P{S;R/m} (38) Clearly the rules Ri{a*, P*) and R/ {a*, P*) are equivalent so that for g > we haA-e P{8;R-s{a*)} ^ P{8;R/ia*)] (39) REDUCING TIME IN RELIABILITY STUDIES 197 and hence, in particular, letting 8 = 8* in (38) we have P{8*;R,(a*)} ^ P{8*;R,"(8*)] ^ P* (40) since the right member of (34) reduces to P* when W is an integer and 5 = 5*. The error in the approximations above can be disregarded when g is small compared to 02 . Thus we have shown that for small values of g/d2 the probability specification based on (a*, ^*, P*) is satisfied in the sense of (40) if we use the procedure Rsia*, P*), i.e., if we proceed as if It would be desirable to show that w^e can proceed as if g = for all values of g and P*. It can be shown that for swfficiently large n the rule Ri{a*, P*) meets it specification for all g. One effect of increasing n is to decrease the average time E{t) between failures and to approach the corresponding problem without replaceme^it since g/E{T) becomes large. Hence we need only show that Ri{a*, P*) meets its specification for the corresponding problem without replacement. If we disregard the information furnished by Poisson life and rely solely on the counting of failures then the problem reduces to testing in a single binomial whether 6 = di for population IIi and 6 = do for population 112 or vice versa. Let- ting p denote the probability that the next failure arises from 111 then we have formally tia'-V = -. — ; — versus Hi-.p = 1 + a ^ 1 + a For preassigned constants a* > I and P* (V2 < P* < 1) the appropri- ate sequential likelihood test to meet the specification: "Probability of a Correct Selection ^ P* whenever a ^ a*" (41) then turns out to be precisely the procedure Rsia*, P*). Hence we may proceed as if gr = when n is sufficiently large. The specifications of the problem may be given in a different form. Suppose 01* > 02* are specified and it is desired to haxe a probability of a correct selection of at least P* whenever ^1 ^ 0i* > 02* ^ 02 . Then we can form the following sequential likelihood procedure R3* which is more efficient than Rsia*, P*). Rule /?3*.- "Continue experimentation without replacement until a time t is reached at which the inequality 198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 is satisfied. Then stop and select the population with ri failures as the population with d = di". It can be easily shown that the greatest lower bound of the bracketed quantity in (42) is 0i*/^2*. Hence for di*/d2* = a* and P* > i 2 the time required by Rz*{6i*, 62*, P*) ivill always be less than the time required by R,(a*,P*). Another type of problem is one in which we are given that 6 = di* for one population and d = 62* for the A; — 1 others where 6]* > 62* are specified. The problem is to select the population with 6 = di*. Then (42) can again be used. In this case the parameter space is discrete with k points only one of which is correct. If Rule R3* is used then the probability of selecting the correct point is at least P*. Equilibrium Approach When Failures Are Replaced 9 Consider first the case in which all items on test are from the same exponential population with parameters (6, g). Let Tnj denote the length of the time interval between the j^^ and the j + 1^* failures, (j = 0, 1, • • • ), where n is the number of items on test and the 0*'' failure de- notes the starting time. As time increases to infinity the expected number of failures per unit time clearly approaches n/(0 + g) which is called the equilibrium failure rate. The inverse of this is the expected time between failures at equilibrium, say E{Tn^). The question as to how the quanti- ties E{Tnj) approach E(Tn^) is of considerable interest in its own right. The following results hold for any fixed integer 71 ^ 1 unless explicitly stated otherwise. It is easy to see that ^^(^i) ^ E{TnJ ^ E(T„o) (43) since the exact values are respectively e /, e-^-^'^/^X ^ g+d ^ , d < ^ 9+ - (44) n — 1 \ n / n n In fact, since all units are new at starting time and since at the time of the first failure all units (except the replacement) have passed their guarantee period with probability one then ^(^i) ^ E(Tnj) S E{Tn,) (j ^ 0) (45) If we compare the case g > with the special case g = we obtain E{2\j) ^ - (y= 1,2, •••) (46) n REDUCING TIME IN RELIABILITY STUDIES 199 and if we compare it with the non-replacement case {g/Q is large) we obtain ^(n,) ^ -^. (i = 1, 2, • . • , n - 1). (47) These comparisons show that the difference in (46) is small when g/0 is small and for j < n the difference in (47) is small when g/d is large. It is possible to compute E{Tnj) exactly for g ^ but the computa- tion is extremely tedious for j ^ 2. The results for j = 1 and are given in (44). Fori = 2 E(Tn2) = n (n + 2)(/i - 1) -(n-2)gie 1 - ' ' ': -e n + Vl^iI g-(«-i)p/^ ri-2_ -un-i),ie I {n>2) n — \ v?{n — 1) and 2{n-l)glB (48) E{T,.^ = ^ - ^ [1 - ^e-'" + e-'"'\ (49) For the case of two populations with a common guarantee period g we can write similar inequalities. We shall use different symbols a, h for the initial sample size from the populations with scale parameters Oi , O2 respectively even though our principal interest is in the case a = b = n say. Let Ta,b.j denote the interval between the j^^ and j -f P* fail- ures in this case and let X, = l/di (i = 1,2). We then have for all values of a and b [aXi + b\o]-' ^ E(TaXj) ^ E(Taxo) = g + [aXi + b\,]-' (j = 0,1,2, ■■■, ^) (50) J?(T ^ (gl + g){e2 + g) .riN a{92 -h 9) + b{di + g) The result for E(Ta,b.i) corresponding to that in (43) does not hold if the ratio di/62 is too large; in particular it can be shown that -0[(a-l)Xi+6X2l-l E{T.,b..) = ^ "^^ ^' ^ aXi + 6X2/ \(a — l)Xi 4- 6X2 _ Xie aXi + 6X2 + / ^X2 Y 1 \r x^e-''^'^^''-''''-'- (52) ,aXi -\- bX2/\aXi + (& — 1)^2 L 0X1 + ^^2 is larger than E{Ta,h.J for a = 6 = 1 when ^/^i = 0.01 and g/di = 0.10 200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 SO that QilQi = 10. The expression (52) reduces to that in (44) if we set di = 02 = 6 and replace a and h by n/2 in the resulting expression. Corresponding exact expressions for E(Ta.b,j) for j > 1 are extremely tedious to derive and unwieldy although the integrations involved are elementary. If we let g —^ oo then we obtain expressions for the non- replacement case which are relatively simple. They are best expressed as a recursion formula. E(.Ta,bj) = — , ,. ETa-\,b,}-l + m^ ^"— ^^ = '^ (53) EiT.,b.d = "^^ ^ aXi + 6X2 (a — l)Xi + 6X2 I 0X2 1 ( h > ^^ "^ aXi + 6X2 aXi + (6 - 1)X2 ' = (54) E(Tafij) ^ g + di/a fori ^ a and j = (55) E{Ta,oJ = dr/(a -j) for 1 ^ i ^ a - 1 (56) Results similar to (55) and (56) hold for the case a = 0. The above results for gr = 00 provide useful approximations for E{Ta,b,j) when g is large. Upper bounds are given by M E{Ta,bj) ^ [aXi + (6 - i)X2r (i = 1, 2, • • • , h) (57) E(Ta.bj+b) ^ [(a - j)Xr' (i = 1, 2, • . • , a - 1). (58) Duration of the Experiment For the sequential rule R^' with k = 2 we can now write down approxi- mations as well as upper and lower bounds to the expected duration E{T) of the experiment. From (50) I g + ..5^;^.\ s E(T) = E /?(r.,,) c-l n(Xi -f X2) ^ '''^ ' ~ § '^^^ "'"'^^ (59) + \FA¥; 5) - c]i!;(T„,„,.) where c is the largest integer less than or equal to E{F\ 5). The right ex- pression of (59) can be approximated by (53) and (54) if g is large. If c < 2n then the upper bounds are given by (57) and (58). A simpler j REDUCING TIME IN RELIABILITY STUDIES 201 upper bound, which holds for all \'aliies of c is given by E{T) ^ E{F- b)E{Tn,n..) = E{F; 8) (g + ^^ (60) CASE 3: COMMON UNKNOWN LOCATION PARAMETER ^ ^ In this case the more conservative procedure is to proceed under the assumption that </ = 0. By the discussion above the probability require- ment will in most problems be satisfied for all ^ ^ 0. The OC and ASN functions, which are now functions of the true value of g, were already obtained above. Of course, we need not consider values of g greater than the smallest observed lifetime of all units tested to failure. Addendum 2 For completeness it would be appropriate to state explicitly some of the formulas used in computing the tables in the early part of the paper. For the nonsequential, nonreplacement rule Ri with /c = 2 the proba- bility of a correct selection is P(a; R,) = [ [ Mu, OAfrix, 6,) dy dx (61) where fXx, e) = '- C(l - e^'"y-' e-^^"-^+^"^ (r ^ n) (62) and C" is the usual combinatorial symbol. This can also be expressed in the form P{a; R,) = 1 - (rC:r Z ^~^^"' ;=i n - r -\-j (63) C'-l{B[r, n-r+l+a(n-r+ j)]}-' where B[x, y] is the complete Beta function. Eciuation (66) holds for any g ^ 0. For the rule Ri the expected duration of the experiment for k = 2 is given by E{T) = r x{fr(x, d,)[l - Frix, 62)] + frix, d,)[l - Fr(x, ^i)] } dx (64) •'0 where frix, 6) is the density in (62) and Fr{x, B) is its c.d.f. This can 202 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 also be expressed in the form ^iKC^ZZt (-1) c.-. c, . plus another similar expression in which 6i , a are replaced by 62 , a~^ respectively. For ^ > we need only add g to this result. This result was used to compute E(T) in table lA f or a = 1 and a = 2. For a = oo the expression simplifies to E{T) = e^rC: ± erl ^~^^'^\ (66) which can be shoAvn to be equivalent to E{T) = e,f: ^— (67) REFERENCES 1. Bechhofer, R. E., Kiefer, J. and Sobel, M., On a Type of Sequential Multiple Decision Procedures for Certain Ranking and Identification Problems with k Populations. To be published. 2. Birnbaum, A., Statistical methods for Poisson processes and exponential populations, J. Am. Stat. Assoc, 49, pp. 254-266, 1954. 3. Birnbaum, A., Some procedures for comparing Poisson processes or popula- tions, Biometrika, 40, pp. 447-49, 1953. 4. Girshick, M. A., Contributions to the theory of sequential analj'sis I, Annals Math. Stat., 17, pp. 123-43, 1946. 5. Wald, A., Sequential Analysis, John Wiley and Sons, New York, 1947. I A Class of Binary Signaling Alphabets By DAVID SLEPIAN (Manuscript received September 27, 1955) A class of binary signaling alphabets called "group alphabets" is de- scribed. The alphabets are generalizations of Hamming^ s error correcting codes and possess the following special features: {1) all letters are treated alike in transmission; {2) the encoding is simple to instrument; (3) maxi- mum likelihood detection is relatively simple to instrument; and (4) in certain practical cases there exist no better alphabets. A compilation is given of group alphabets of length equal to or less than 10 binary digits. INTRODUCTION This paper is concerned with a class of signahng alphabets, called "group alphabets," for use on the symmetric binary channel. The class in question is sufficiently broad to include the error correcting codes of Hamming,^ the Reed-Muller codes," and all "systematic codes''.^ On the other hand, because they constitute a rather small subclass of the class of all binary alphabets, group alphabets possess many important special features of practical interest. In particular, (1) all letters of the alphabets are treated alike under transmission; (2) the encoding scheme is particularly simple to instru- ment; (3) the decoder — a maximum likelihood detector — is the best I possible theoretically and is relatively easy to instrument; and (4) in certain cases of practical interest the alphabets are the best possible theoretically. It has very recently been proved by Peter Elias^ that there exist group alphabets which signal at a rate arbitarily close to the capacity, C, of the symmetric binary channel with an arbitrarily small probability of error. Elias' demonstration is an existence proof in that it does not show explicitly how to construct a group alphabet signaling at a rate greater than C — e with a probability of error less than 5 for arbitrary positive 5 and e. Unfortunately, in this respect and in many others, our understanding of group alphabets is still fragmentary. In Part I, group alphabets are defined along with some related con- 203 204 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 cepts necessary for their understanding. The main results obtained up to the present time are stated without proof. Examples of these concepts are given and a compilation of the best group alphabets of small size is presented and explained. This section is intended for the casual reader. In Part II, proofs of the statements of Part I are given along with such theory as is needed for these proofs. The reader is assumed to be familiar with the paper of Hamming, the basic papers of Shannon* and the most elementary notions of the theory of finite groups. Part I — Group Alphabets and Their Properties 1.1 INTRODUCTION We shall be concerned in all that follows with communication over the symmetric binary channel shown on Fig. 1. The channel can accept either of the two symbols or 1 . A transmitted is received as a with probability q and is received as a 1 w'ith probability p — 1 — g : a trans- mitted 1 is received as a 1 with probability q and is received as a with probability p. We assume ^ p ^ ^^. The "noise" on the channel operates independently on each symbol presented for transmission. The capacity of this channel is C = 1 + P log2P + q log29 bits/symbol (1) By a K-leUer, n-place binary signaling alphabet we shall mean a collec- tion of K distinct sequences of n binary digits. An individual sequence of the collection will be referred to as a letter of the alphabet. The integer K is called the size of the alphabet. A letter is transmitted over the channel by presenting in order to the channel input the sequence of n zeros and ones that comprise the letter. A detection scheme or detector for INPUT X OUTPUT Fig. 1 — The symmetric binary channel. A CLASS OF BINARY SIGNALING ALPHABETS 205 a given /v-letter, n-place alphabet is a procedure for producing a sequence of letters of the alphabet from the channel output. Throughout this paper we shall assume that signaling is accomplished with a given /i-letter, n-place alphabet by choosing the letters of the alphabet for transmission independently with equal probability l/K. Shannon^ has shown that for sufficiently large n, there exist K-letter, n-place alphabets and detection schemes that signal over the symmetric binary chaimel at a rate R > C — e for arbitrary £ > and such that the probability of error in the letters of the detector output is less than any 5 > 0. Here C is given by (1) and is shown as a function of p in Fig. 2. No algorithm is known (other than exhaustvie procedures) for the construction of A'-letter, /i-place alphabets satisfying the above inequalities for arbitrary positive 8 and e except in the trivial cases C — and C = 1. 1.2 THE GROUP -S„ There are a totality of 2" different w-place binary sequences. It is fre- quently convenient to consider these sequences as the vertices of a cube of unit edge in a Euclidean space of n-dimensions. For example the 5- place sequence 0, 1, 0, 0, 1 is associated with the point in 5-space whose o.e 0.6 0.4 0.2 Fig. 2 — The capacity of the symmetric binary channel. C = 1 + p log2 p + {I - p) log2 (1 - p) 206 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 coordinates are (0, 1, 0, 0, 1). For convenience of notation we shall gen- erally omit commas in writing a sequence. The above 5-place sequence will be written, for example, 01001. We define the product of two n-ylace hinarij sequences, aicii • • • a„ and ^1^2 • ■ • bn as the n-place binary sequence fli + hi , a-i ■]- h-i , ■ ■ • , ttn + hn Here the a's and 6's are zero or one and the + sign means addition modulo 2. (That is + 0=1 + 1 = 0, 0+1 = 1+0=1) For example, (01101) (00111) = 01010. With this rule of multiplication the 2" w-place binary sequences form an Abelian group of order 2". The elements of the group, denoted by Ti , T'2 , • • • , Tin, say, are the n-place binary sequences ; the identity element I is the sequence 000 • • • and IT, = Til = T. ■ T,Tj = TjTr, TiiTjT,) = iTiTj)Tk ; the product of any number of elements is again an element; every ele- ment is its own reciprocal, Ti = Tf^, TI = /. We denote this group by Bn . All subgroups of Bn are of order 2 where k is an integer from the set 0, 1, 2, • • • , n. There are exactly N{n, k) = (2" - 2") (2" - 2') (2" - 2') • • • (2" - 2'-') (2^ - 2»)(2'^ - 20(2* - 22) = N(n, n — k) {2" - 2'-') (2) distinct subgroups of Bn of order 2 . Some values of N(n, k) are given in Table I. Table I — Some Values of A^(n, k), the Number of Subgroups OF Bn OF Order 2''. N(n, k) = N{n, n — k) n\k 1 2 3 4 5 2 3 1 3 7 7 1 4 15 35 15 1 5 31 155 155 31 1 6 63 651 1395 651 63 7 127 2667 IISU 11811 2667 8 255 10795 97155 200787 97155 9 511 43435 788035 3309747 3309747 10 1023 174251 6347715 53743987 109221651 000 000 000 000 000 000 000 100 100 100 010 010 001 no 010 001 oil 001 101 no on 110 101 111 on 111 111 101 A CLASS OF BINARY SIGNALING ALPHABETS 207 1.3 GROUP ALPHABETS An ?i-place group alphabet is a 7v-letter, n-place binary signaling alpha- bet whose letters form a subgroup of Bn . Of necessity the size of an n-place group alphabet is /v = 2 where k is an integer satisfying ^ k ^ n. By an (n, k)-alphahet we shall mean an n-place group alphabet of size 2^. Example: the N{3, 2) = 7 distinct (3, 2)-alphabets are given by the seven columns (i) (ii) (iii) (iv) (v) (vi) (vii) (3) 1.4 STANDARD ARRAYS Let the letters of a specific (n, /i:)-alphabet be Ai = / = 00 • • • 0, Ao , As , • ■ ■ , A^ , where ju = 2 . The group Bn can be developed accord- ing to this subgroup and its cosets: /, A2, A3, ■■• ,A^ S2 , S2A2 , S2A3 , • • • , S2A^ Sz , S3A2 , S3A3 , • • • , SsA^ Bn = ; (4) Sr f SyA2 , SpAz , • ' • , SfAfi In this array every element of Bn appears once and only once. The col- lection of elements in any row of this array is called a coset of the (n, k)- alphabet. Here *S2 is any element of B„ not in the first row of the array, S3 is any element of Bn not in the first two rows of the array, etc. The elements S2 , S3 , • • • , Sy appearing under I in such an array will be called the coset leaders. If a coset leader is replaced by any element in the coset, the same coset will result. That is to say the two collections of elements Si , ^1^2 , SiSz ; ■ • ■ , SiA^ and SiA,, , (SiAu)A2 , (SiAMs ,■■■ {SiAk)A, are the same. 208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G We define the weight Wi = w{Ti) of an element, Ti , of Bn to be the number of ones in the n-place binary sequence T,- . Henceforth, unless otherwise stated, we agree in dealing with an ar- ray such as (4) to adopt the following convention: the leader of each coset shall be taken to be an . . element of minimal weight in that coset. Such a table will be called a standard array. Example: Bi can be developed according to the (4, 2)-alphabet 0000, 1100, 0011, nil as follows (6) 0000 1100 0011 nil 1010 Olio 1001 0101 1110 0010 1101 0001 1000 0100 1011 0111 )W"ever, ^^-e should write. for exan 0000 1100 0011 nil 1010 0110 1001 0101 0010 1110 0001 1101 1000 0100 1011 0111 (7) The coset leader of the second coset of (6) can be taken as any element of that row since all are of weight 2. The leader of the third coset, how- ever, should be either 0010 or 0001 since these are of weight one. The leader of the fourth coset should be either 1000 or 0100. 1.5 THE DETECTION SCHEME Consider now communicating with an (n, fc) -alphabet over the sym- metric binary channel. When any letter, say A,, of the alphabet is transmitted, the received sequence can be of any element of B„ . We agree to use the following detector: if the received element of Bn lies in column i of the array (4), the detector prints the letter Ai ,i = 1,2, • • • , ju. The array (4) is to (8) be constructed according to the convention (5). The following propositions and theorems can be proved concerning signaling with an (n, /c)-alphabet and the detection scheme given by (8). 1.6 BEST DETECTOR AND SYMMETRIC SIGNALING Define the probability /,• = ((Ti) of an element Ti of Bn to be A = ^wi^n-uf ^yYiere p and q are as in (1) and Wi is the weight of Ti . Let A CLASS OF BINARY SIGNALING ALPHABETS 209 Qi , i = 1 , 2, • • • , jLi be the sum of the probabilities of the elements in the iih. column of the standard array (4). Proposition 1. The probability that any transmitted letter of the (n, A;) -alphabet be produced correctly by the detector is Qi . Proposition 2. The equivocation^ per symbol is 1 ** Hy{x) = — S Qi log2 Qi n i=i Theorem 1 . The detector (8) is a maximum likelihood detector. That is, for the given alphabet no other detection scheme has a greater average probability that a transmitted letter be produced correctly by the de- tector. Let us return to the geometrical picture of w-place binary sequences as vertices of a unit cube in n-space. The choice of a i^-letter, n-place alphabet corresponds to designating K particular vertices as letters. Since the binary sequence corresponding to any vertex can be produced by the channel output, any detector must consist of a set of rules that associates various vertices of the cube with the vertices designated as letters of the alphabet. We assume that every vertex is associated with some letter. The vertices of the cube are divided then into disjoint sets, Wi , Wi , • • • , Wk where Wi is the set of vertices associated with tth letter of the signaling alphabet. A maximum likelihood detector is char- acterized by the fact that every vertex in Wi is as close to or closer to the iih. letter than to any other letter, i = 1,2, • • • , K. For group alpha- bets and the detector (8), this means that no element in the iih. column of array (4) is closer to any other A than it is to ^i , z = 1, 2, • • • , ;u. Theorem 2. Associated with each {n, /(;)-alphabet considered as a point configuration in Euclidean n-space, there is a group of n X n orthogonal matrices which is transitive on the letters of the alphabet and which leaves the unit cube invariant. The maximum likelihood sets 1^1 , W2 , • • • Wn are all geometrically similar. Stated in loose terms, this theorem asserts that in an (n, A;)-alphabet every letter is treated the same. Every two letters have the same number of nearest neighbors associated with them, the same number of next nearest neighbors, etc. The disposition of points in any two W regions is the same. 1.7 GROUP ALPHABETS AND PARITY CHECKS Theorem 3. Every group alphabet is a systematic^ code: every syste- matic code is a group alphabet. 210 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We prefer to use the word "alphabet" in place of "code" since the latter has many meanings. In a systematic alphabet, the places in any letter can be divided into two classes : the information places — A; in number for an (n, /c)-alphabet — and the check positions. All letters have the same information places and the same check places. If there are k information places, these may be occupied by any of the 2 /v-place binary sequences. The entries in the n — k check positions are fixed linear (mod 2) combinations of the entries in the information positions. The rules by which the entries in the check places are determined are called parity checks. Examples: for the (4, 2)-alphabet of (6), namely 0000, 1100, 0011, nil, positions 2 and 3 can be regarded as the informa- tion positions. If a letter of the alphabet is the sequence aia^a^ai , then ai = a2 , tti = az are the parity checks determining the check places 1 and 4. For the (5, 3)-alphabet 00000, 10001, 01011, 00111, 11010, 10110, 01100, 11101 places 1, 2, and 3 (numbered from the left) can be taken as the information places. If a general letter of the alphabet is aiazazaiai , then a4 = a2 -j- as , Ob = ai -j- a2 -|- ^3 . Two group alphabets are called equivalent if one can be obtained from the other by a permutation of places. Example: the 7 distinct (3, 2)- alphabets given in (3) separate into three equivalence classes. Alpha- bets (i), (ii), and (iv) are equivalent; alphabets (iii), (v), (vi), are equiva- lent; (vii) is in a class by itself. Proposition S. Equivalent (n, fc) -alphabets have the same probability Qi of correct transmission for each letter. Proposition 4- Every (n, /c) -alphabet is equivalent to an (n, k)- alphabet whose first k places are information places and whose last n — k places are determined by parity checks over the first k places. Henceforth we shall be concerned only with (n. A;) -alphabets w^hose first k places are information places. The parity check rules can then be written k ai = S Tij-ay , t = /b -j- 1, • • • , n (9) where the sums are of course mod 2. Here, as before, a typical letter of the alphabet is the sequence aia^ • ■ - ttn . The jn are k(n — k) quantities, zero or one, that serve to define the particular (n, A;)-alphabet in question. 1.8 MAXIMUM LIKELIHOOD DETECTION BY PARITY CHECKS For any element, J\ of Bn we can form the sum given on the right of (9). This sum maj^ or may not agree with the symbol in the ?'th place of A CLASS OF BINARY SIGNALING ALPHABETS 211 T. If it does, we say T satisfies the tth-place parity check; otherwise T fails the zth-place parity check. When a set of parity check rules (9) is giN'cii, we can associate an (n — /i^-place binary sequence, R{T), with each element T of 5„. We examine each check place of T in order starting with the (k -\- 1 )-st place of T. We write a zero if a place of T satisfies the parity check; we write a one if a place fails the parity check. The re- sultant sequence of zeros and ones, written from left to right is R(T). We call R(T) the parity check sequence of T. Example: with the parity rules 04 = 02 -j- 03 , 05 = Oi -j- 02 -j- c^s used to define the (5, 3)-alphabet in the examples of Theorem 3, we find i?(11000) = 10 since the sum of the entries in the second and third places of 11001 is not the entry of the fourth place and since the sum of Oi = 1, 02 = 1, and 03 = is = 05 . Theorem 4- Let I, A2 , • • • ^^^ be an {n, /c)-alphabet. Let R{T) be the parity check sequence of an element T of B„ formed in accordance with the parity check rules of the (n, /c) -alphabet. Then R(Ti) = R(T2) if and only if Ti and T2 lie in the same row of array (4). The coset leaders can be ordered so that R{Si) is the binary symbol for the integer i — 1. As an example of Theorem 4 consider the (4, 2)-alphabet shown with its cosets below 0000 1011 0101 1110 0100 nil 0001 1010 0010 1001 0111 1100 1000 0011 1101 0110 The parity check rules for this alphabet are 03 = oi , 04 = Oi -j- ^2 • Every element of the second row of this array satisfies the parity check in the third place and fails the parity check in the 4th place. The parity check sequence for the second row is 01. The parity check for the third row is 10, and for the fourth row 11. Since every letter of the alphabet satisfies the parity checks, the parity check sequence for the first row is 00. We therefore make the following association between parity check sequences and coset leaders 00 -^ 0000 = Si 01 -^ 0100 = S2 10 -^ 0010 = S, 11 -^ 1000 = ^4 1.9 INSTRUMENTING A GROUP ALPHABET Proposition 4 attests to the ease of the encoding operation involved 212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 with the use of an (n, fc) -alphabet. If the original message is presented as a long sequence of zeros and ones, the sequence is broken into blocks of length k places. Each block is used as the first k places of a letter of the signaling alphabet. The last n-k places of the letter are determined by fixed parity checks over the first k places. Theorem 4 demonstrates the relative ease of instrumenting the maxi- mum hkelihood detector (8) for use with an (n. A:) -alphabet. When an element T of Bn is received at the channel output, it is subjected to the n-k parity checks of the alphabet being used. This results in a parity check sequence R{T). R(T) serves to identify a unique coset leader, say Si . The product SiT is then formed and produced as the detector out- put. The probability that this be the correct letter of the alphabet is Qi . 1.10 BEST GROUP ALPHABETS Two important questions regarding (n, fc)-alphabets naturally arise. What is the maximum value of Qi possible for a given n and k and which of the N(n, k) different subgroups give rise to this maximum Qi? The answers to these questions for general n and k are not known. For many special values of n and k the answers are known. They are presented in Tables II, III and IV, which are explained below. The probability Qi that a transmitted letter be produced correctly by the detector is the sum, Qi = ^i f{Si) of the probabilities of the coset leaders. This sum can be rewritten as Qi = 2Zi=o «« P^Q^~^ where a, is the number of coset leaders of weight i. One has, of course, ^a, = v = / y) \ T? ' 2^"'' for an (n, /(;)-alphabet. Also «> ^ ( . ) = -7-7 — '■ — n- ! since this is the \t / tlin — t) number of elements of Bn of weight i. The (Xi have a special physical significance. Due to the noise on the channel, a transmitted letter, A, , of an (n, /c)-alphabet will in general be received at the channel output as some element T of Bn different from Ai .li T differs from Ai in s places, i.e., if w{AiT) = s, we say that an s-tuple error has occurred. For a given (n, fc)-alphabet, ai is the number of i-tuple errors which can be corrected by the alphabet in question, i = 0, 1,2, ■ • • , n. Table II gives the a{ corresponding to the largest possible value of Qi for a given k and ?i for k = 2,3, •••w— l,n = 4--- ,10 along with a few other scattered values of n and k. For reference the binomial coeffi- cients ( . ) are also listed. For example, we find from Table II that the best group alphabet with 2 =16 letters that uses n = 10 places has a A CLASS OF BINARY SIGNALING ALPHABETS 213 1 A Q C 'J ** Q probability of correct transmission Qi = q + lOg p + 39g p" + l-Ag'p . The alphabet corrects all 10 possible single errors. It corrects 39 of the possible f .^ j = 45 double errors (second column of Table II) and in addition corrects 14 of the 120 possible triple errors. By adding an addi- tional place to the alphabet one obtains with the best (11, 4)-alphabet an alphabet with 16 letters that corrects all 11 possible single errors and all 55 possible double errors as well as 61 triple errors. Such an alphabet might be useful in a computer representing decimal numbers in binary form. For each set of a's listed in Table II, there is in Table III a set of parity check rules which determines an {n, A)-alphabet having the given a's. The notation used in Table III is best explained by an example. A (10, 4)-alphabet which realizes the a's discussed in the preceding para- graph can be obtained as follows. Places 1, 2, 3, 4 carrj- the information. Place 5 is determined to make the mod 2 sum of the entries in places 3, 4, and 5 ecjual to zero. Place 6 is determined by a similar parity check on places 1, 2, 3, and 6; place 7 by a check on places 1, 2, 4, and 7, etc. It is a surprising fact that for all cases investigated thus far an {n, k)- alphabet best for a given value of p is uniformly best for all values of p, ^ p ^ 1 2. It is of course conjectured that this is true for all n and /,-. It is a further (perhaps) surprising fact that the best {n, fc) -alphabets are not necessarily those with greatest nearest neighbor distance be- tween letters when the alphabets are regarded as point configurations on the n-cube. For example, in the best (7, 3)-alphabet as listed in Table III, each letter has two nearest neighbors distant 3 edges away. On the other hand, in the (7, 3)-alphabet given by the parity check rules 413, 512, 623, 7123 each letter has its nearest neighbors 4 edges away. This latter alphabet does not have as large a value of Qi , however, as does the (7, 3)-alphabet listed on Table III. The cases /.; = 0, 1, /? — 1, n have not been listed in Tables II and III. The cases k = and k = n are completely trivial. For k = 1, all n > 1 the best alphabet is obtained using the parity rule a> = 03= • • • = a„ = oi . If n = '2j, If n = 2j + 1, Qi = i: (^') pY-\ For k = n — 1, /; > 1. the maximum Qi is Qi = g"~ and a parity rule for an alphabet realizing this Qi is o„ = oi . If the a's of an (/<, A)-alphabet are of the form a, = ( . j , i = 0, 1, 214 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table II — Probability of No Error with Best Alphabets, Qi = 2Z «»P*2"~' (?) k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 * = 10 i ai (li a a, ai Oi fli ai a; n = 4 1 1 1 4 3 1 1 1 n = 5 1 2 5 10 1 5 2 1 3 1 1 71 = 6 1 2 6 15 6 9 6 1 3 1 1 1 1 1 n = 7 1 2 3 7 21 25 7 18 6 7 8 7 3 1 1 1 1 1 1 n = 8 1 2 3 8 28 56 8 28 27 8 20 3 8 7 7 3 1 1 1 1 1 1 1 1 9 9 9 9 9 7 3 n = 9 2 3 4 36 84 126 36 64 18 33 21 22 6 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 7 3 n = 10 2 3 4 45 120 210 45 110 90 45 64 8 39 14 21 5 1 1 1 1 1 1 1 1 1 11 11 11 11 11 11 7 3 n = 11 2 3 4 5 55 165 330 462 55 165 226 54 55 126 63 55 61 20 4 1 1 1 1 1 1 1 1 12 12 12 12 12 7 3 n = 12 2 3 4 5 66 220 495 792 66 220 425 300 66 200 233 19 3 A CLASS OF BINARY SIGNALIHG ALPHABETS 215 2, • • • , j, «j+i = f some integer, aj+o = ay+s = • • • = «„ = 0, then there does not exist a 2 -letter, w-place alphabet of any sort better than the given (n, A)-alphabet. It will be observed that many of the a's of Table II are of this form. It can be shown that Proposition 5 ii n -\- I „ /"t"! q 1^2"^* — 1 there exists no 2'''-letter, n-place alphabet better than the best (n, /c) -alphabet. When the inequality of proposition 5 holds the a's are either «o = 1, ""'' - 1, all other « = 0; or ao = 1, «i = (Vj , «2 = 2"~' - 1 - , all other a = 0; or the trivial ao = 1 all other a = which holds uhen k = n. The region of the n — k plane for which it is known that (n, A-)-alphabets cannot be excelled by any other is shown in Table IV. 1.11 A DETAILED EXAMPLE As an example of the use of {n, A") -alphabets consider the not un- realistic case of a channel with -p = 0.001, i.e., on the average one binary digit per thousand is received incorrectly. Suppose we wish to transmit messages using 32 different letters. If we encode the letters into the 32 5-place binary sequences and transmit these sequences without further encoding, the probability that a received letter be in error is 1 — (1 _ pf = 0.00449. If the best (10, 5)-alphabet as shown in Tables II and III is used, the probability that a letter be wrong is 1 — Qi = 1 - r/" - lOgV - 21gy - 24/)' - 72p' + • • • = 0.000024. Thus by reducing the signaling rate by ^^, a more than one hundredfold re- duction in probability of error is accomplished. A (10, 5)-alphabet to achieve these results is given in Table III. Let a typical letter of the alphabet be the 10-place sequence of binary digits aia2 ■ • • agttio . The symbols aia^Ozaia^ carry the information and can be any of 32 different arrangements of zeros and ones. The remaining places are determined by 06 = ai -j- a-i -j- a4 -j- ^5 a? = tti -j- oo -f a4 -j- as as = ai -j- a2 + a.3 + Os ag = Oi + 02 4- Qi -j- 0,4 Oio = Oi + a-i -j- 03 4- 04 4- «5 To design the detector for this alphabet, it is first necessary to deter- mine the coset leaders for a standard array (4) formed for this alphabet. •Jl t-l a pa < M Ph < cc o H O H ti; O H I— I -< Ph P3 t^ 00 -f ^ cc CC C^) !M O t^ X lO a; t^ oc 00 C2 ^ ^ CC CC (N C^l t- GC lO ic lO -r -f -^ CC CT CC C^ CM C^I ;C 1^ X c: ^ cc -+ -f -^ cc -f -^ cc cc -r -^ cc re cc (M <N CC C^ CM CC CM CM CM re T-l CM CM CM ic :c I- y: — i re cc CO ce C^l cc CM cc re re C^J CM CM CM re re c^i CM re re CM ^— .-H .-H -— C^l _ ,_ — ,-H r— ^- T-H (M ,— . T-^ CM ^ ^ CM -^ '^^ lo •^ lO « -* iC <£) t^ •^ lO CO t^ oc "* >OCD t^OO C5 C^l ex re C^l CM C^l re-rocot^ ce-^iocot^oc C^l C^l ' >o re f lO CO re T lO CO t^ oC' iCi CO oc 210 1—1 1—1 ^CM 1-H 1-H cO'f -* CM CM CO 1-1 1-l T— 1 1-H 1-H Ot-h Oi-HCM 1—1 1—1 I-H 1-H 1-H 00 ^cot-oo ^^iCiO 134 124 1 123 12351 123 1 124 2134 Ol ^ 01 .—1 1—1 "^ 1-H 1-H 1-H r^ t- t- t^ coco CO CO lO •^iO-* ^^10^-^ CO "^^00 CO "''^ CO CO CO -* 't< jvj ^'*(N^ '^'* CM CM CM CO(M^ «^^^ COCM^^^ ^-^o -^-^O-H ^'~' 0--HCM GOO-J T-1 00 01 1-1 1—1 00 02 1-H 1-H 1-H CO CO iC ^-t OCOCO^^ iOiO>Oj^ '^'^'^coco -^-^^0^ '^'^^cacM CO!M C^ ^ CO <M CM ^ ^ T— I 1-H T-H _^ 1— 1 1 — 1 r— * -^ o C "—1 t^ 00 o i-H t^ 00 C5 1— 1 >— 1 iCi 'I* lOiOiO'*^ j^ •r-^ c^cc ^ CO CM iM O) ^ CO t^ 00 02 ^ •* ^ CO ^co CO^ ^ -* CM '^'cOCM 'f CM (M CO CO CO ^ "* coco j^ 1— H CO T-H 1— 1 ^H C^l 1— 1 CM r-( CM .-< ^ 1— 1 10 CO t^ 00 cr. 10 CO t^ 00 rH 1-H CO coco CO CO CM (N (M CO CM CM CI CO CO r— 1 coco CM T—l T~i o*o*c<i^^^ 1— 1 CM CO r- 1 .— 1 CM 1— ( CO CO CM 1-1 1—1 1—1 1-H ^ CM CO -1 1-H -H ^ ^ ^ ^lOfOi^ccoi •^ lO CO t^ 00 Ci I— 1 1— 1 rflOCOt^OOOli-Hi-Hi-H CM C^ C^l CM CM CM CM (M !M T-H 1—1 1— t t— 1 1 — I 1— 1 ^H I— 1 CM CM 1-H T— t r-i .-< i-H CM CM CM 1— t 1-H ^ ^ ^ ^ CM CM CM ^ ^ ^ CO 'tl lO CO t^ 00 05 >— 1 CO ■* 10 CO t> 00 Oi i-H r— ( CO-*lOCOt^00C2l-Hr-l^ I-l CM T— 1 1— ( 1— t II II II e e e 217 218 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table IV — Region of the n-k Plane for Which it is Known THAT [n, fc)-ALPHABETS CaNNOT Be EXCELLED k 30 29 28 • • • 27 .... 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 \ 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 n This can be done by a \'ariety of special methods which considerably reduce the obvious labor of making such an array. A set of best »S's along with their parity check symbols is given in Table V. A maximum likelihood detector for the (10, 5)-alphabet in question forms from each received sequence 6162 • • • &10 the parity check symbol C1C2C3C4C5 where Ci = h 4- ^h 4- ^3 + Ih + ^5 C2 = 67 -(- 6i -]- h-i + hi \- Ih Cs = &8 + ^^1 + h -j- Ih + ^5 Ci = bg + hi 4- h-i -i- h-.i -(- hi C5 = />in + hi + />, + h, 4- hi 4- 65 According to Table V, if CiC-jCiAf'b contains less than three ones, the de- tector should brint hih^kihih^ . The detector should piint (/m 4- 1)^2^3^4'':. if the parity check sequence C1C2C3C4C5 is either 11111 oi- 11110; the dv- A CLASS OF BINARY SIGNALING ALPHABETS 219 Table V — Coset Leaders and Parity Check Sequences FOR (10, 5) -Alphabet ClCiCsCiCb ^ s CIC2C3C4C6 5 00000 0000000000 11100 0000100001 10000 0000010000 11010 0001000001 01000 0000001000 11001 0001000010 00100 0000000100 10110 0010000001 00010 0000000010 10101 0010000010 00001 0000000001 10011 OOIOOOOIOO 1 1000 0000011000 OHIO 0100000001 10100 0000010100 01101 0100000010 10010 0000010010 01011 0100000100 10001 0000010001 00111 0100001000 01100 0000001100 11110 1000000001 01010 0000001010 11101 OOOOIOOOOO 01001 0000001001 11011 OOOIOOOOOO 00110 0000000110 10111 0010000000 00101 0000000101 01111 0100000000 00011 0000000011 mil 1000000000 tector should print 61(62 -j- l)b3lhh^ if the parity check sequence is 01111, 00111, 01011, 01101, or OHIO; the detector should print hMb-i + 1)6465 if the parity check sequence is 10111, 10011, 10101, or 10110; the de- tector should print 616263(64 -j- 1)65 if the parity check sequence is 11011, 11001, 11010; and finally the detector should print 61626364(65 -j- 1) if the parity check sequence is 11101 or 11100. Simpler rules of operation for the detector may possibly be obtained by choice of a different set of S's in Table V. These quantities in general are not unique. Also there may exist non-equivalent alphabets with simpler detector rules that achieve the same probability of error as the alphabet in question. I'vrt II — Additional Theory and Proofs of Theorems of Part I ' 2.1 the abstract group Cn It will be helpful here to say a few more words about Br, , the group of n-place binary sequences under the operation of addition mod 2. This j group is simply isomorphic with the abstract group Cn generated by n \ commuting elements of order two, say ai, a-2 , ■ ■ ■ , a„ . Here a,:ay = <i,ai and a/ = /, i, j = 1, 2, • • • , n, where / is the identity for the group. The eight distinct elements of C3 are, for example, /, o-i , a-y , (h , (iici-, , aio-.i , a-itti , aia-ittz . The group C„ is easily seen to be isomorphic I with the Ai-fold direct product of the group Ci with itself. 220 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 It is a considerable saving in notation in dealing with C„ to omit the symbol "a" and write only the subscripts. In this notation for example, the elements of d are 7, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234. The product of two or more elements of C„ can readily be written down. Its symbol consists of those numerals that occur an odd number of times in the collection of numerals that comprise the sym- bols of the factors. Thus, (12)(234)(123) = 24. The isomorphism between Cn and Bn can be established in many ways. The most convenient way, perhaps, is to associate with the element iii-2H ■ ■ ■ ik of Cn the element of Bn that has ones in places ii ,1-2, • • • , ik and zeros in the remaining n — k places. For example, one can associate 124 of C4 with 1101 of Bi ; 14 with 1001, etc. In fact, the numeral no- tation afforded by this isomorphism is a much neater notation for Bn than is afforded by the awkward strings of zeros and ones. There are, of course, other ways in which elements of C„ can be paired with elements of Bn so that group multiplication is preserved. The collection of all such "pairings" makes up the group of automorphisms of C„ . This group of automorphisms of Cn is isomorphic with the group of non-singular linear homogenous transformations in a field of characteristic 2. An element T of C„ is said to be dependent upon the set of elements Ti , T2 , • • ■ , Tj oi Cn if T can be expressed as a product of some ele- ments of the set Ti , T2 , • • • , Tj ; otherwise, T is said to be independent of the set. A set of elements is said to be independent if no member can be expressed solely in terms of the other members of the set. For example, in Cs , 1, 2, 3, 4 form a set of independent elements as do likewise 2357, 12357, 14. However, 135 depends upon 145, 3457, 57 since 135 = (145) (3457) (57). Clearly any set of n independent elements of Cn can be taken as generators for the group. For example, all possible products formed of 12, 123, and 23 yield the elements of C3 . Any k independent elements of C„ serve as generators for a subgroup of order 2*". The subgroup so generated is clearly isomorphic with Ck ■ All subgroups of C„ of order 2'' can be obtained in this way. The number of ways in which k independent elements can be chosen from the 2" elements of C„ is F{n, k) - (2" - 2'')(2" - 2')(2" - 2') • • • (2" - 2'-') For, the first element can be chosen in 2" — 1 ways (the identity cannot be included in a non-trivial set of independent elements) and the second element can be chosen in 2" — 2 ways. These two elements determine a subgroup of order 2\ The third element can be chosen as any element of the remaining 2" — 2" elements. The 3 elements chosen determine a I A CLASS OF BINARY SIGNALING ALPHABETS 221 subgroup of order 2l A fourth independent element can be chosen as any of the remaining 2" — 2 elements, etc. Each set of k independent elements serves to generate a subgroup of order 2''. The quantity F{n, k) is not, however, the number of distinct subgroups of C„ of this order, for, a given subgroup can be obtained from many different sets of generators. Indeed, the number of different sets of generators that can generate a given subgroup of order 2^ of C„ is just F{k, k) since any such subgroup is isomorphic with Ck . Therefore the number of subgroups of Cn of order 2'' is N{n, k) = F(n, k)/F(k, k) which is (2). A simple calculation gives N(n, k) = N(n, n — k). 2.2 PROOF OF PROPOSITIONS 1 AND 2 After an element A of 5„ has been presented for transmission over a noisy binary channel, an element T of 5„ is produced at the channel output. The element U = AT oi Bn serves as a record of the noise during the transmission. U is an n-place binary sequence with a one at each place altered in A by the noise. The channel output, T, is obtained from the input A by multiplication by U: T = UA. For channels of the sort under consideration here, the probability that U be any particular element of Bn of w^eight w is p^'g"""'. Consider now signaling with a particular (n, /b) -alphabet and consider the standard array (4) of the alphabet. If the detection scheme (8) is used, a transmitted letter A i will be produced without error if and only if the received symbol is of the form SjAi . That is, there will be no error only if the noise in the channel during the transmission of Ai is represented by one of the coset leaders. (This applies (or i = 1,2, • • • , fi = 2 ). The probability of this event is Qi (Proposition 1, Section 1.6). The convention (5) makes Qi as large as is possible for the given alpha- bet. Let X refer to transmitted letters and let Y refer to letters produced by the detector. We use a vertical bar to denote conditions when writing probabilities. The quantity to the right of the bar is the condition. We suppose the letters of the alphabet to be chosen independently with ec^ual probability 2" . The equivocation h{X \ Y) obtained when using an (n, fc)-alphabet with the detector (8) can most easily be computed from the formula h(X I F) = h{X) - h(Y) + h(Y I X) (10) The entropy of the source is /i(X) = k/n bits per symbol. The probability that the detector produce Aj when Ai was sent is the probability that the noise be represented by AiAjSt , ^ = 1,2, • • • , v. In symbols, 222 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Pr{Y -. Ai I X -^ Ad = Z Pr{N -^ AiA.Sc) = QiA^A,) where Q{Ai) is the sum of the prol)abiUties of the elements that are in the same column as Ai in the standard array. Therefore Pr{Y -> .4,) = E Pr{Y -> A, \ X -^ AdPr{X -^ A^ = ^ E QU,A,) = 4, since E Q^A.A^ = E QUi) = 1. This last follows from the group property of the alphabet. Therefore /i(lO = -- E P>iy -^ A,) log Pr{Y -^ A,) = - bits/symbol. n n It follows then from (10) that h{X I Y) = h(Y I X) The computation of h(Y \ X) follows readily from its definition h{Y I X) = E Prix -^ AdhiY \ X -^ Ai) i = -E Prix -> AdPriY -^ Aj \ X -> Ai) log PHY -^Aj I X-^Ai) = -^,1211 PriN ->AiScAj) log E PriN -> AiS„,Aj) I = -^,ZQiAiAj)'}ogQiAiAj) Zi ij = - EQU,)logQ(A,) i Each letter is n binary places. Proposition 2, then follows. 2.3 DISTANCE AND THE PROOF OF THEOREM 1 Let A and B be two elements of Bn ■ We define the distance, diA, B), between A and B to be the weight of their product, d{A, B) = w(AB) (11) The distance between .4 and B is the number of places in which A and B difTer and is jnsl the "Hamming distance." ^ In terms of the n-cube, diA, B) is Ihe minimum mmiber of edges that must be traversed to go A CLASS OF BINARY SIGNALING ALPHABETS 223 from vertex ^4 to vertex B. The distance so defined is a monotone fnne- tion of the Euchdean distance between vertices. It follows from (11) that if C is any element of B„ then d{A,B) = cJ(A(\BC) (12) This fact shows the detection scheme (8) to be a maximum likelihood detector. By definition of a standard array, one has d(Si , I) ^ d(S,Aj , I) for all i and j The coset leaders were chosen to make this true. From (12), d(S, , I) = d(SiA,„S,- , / .4„.^S,) = d(SiA,n , A,„) d(SAj , /) - diS^AjSiAm , I SiAJ = diAjA,n , SiAr.) = d{SiAm , A() where Af = AjA^ . Substituting these expressions in the inecjuality above yields d(SiAm , A„,) ^ d(SiAm , At) for all i, m, I This equation says that an arbitrary element in the array (4) is at least as close to the element at the top of its column as it is to any other letter of the alphabet. This is the maximum likelihood property. 2.4 PROOF OF THEOREM 2 Again consider an (n, /c) -alphabet as a set of vertices of the unit n-cube. Consider also n mutually perpendicular hyperplanes through the cen- troid of the cube parallel to the coordinate planes. We call these planes "symmetr}^ planes of the cube" and suppose the planes numbered in accordance with the corresponding parallel coordinate planes. The reflection of the vertex with coordinates (ai , a^ , • • • , a^ , • • • , a,j) in symmetry plane i yields the vertex of the cube whose coordinates are (ai , oo , ■ • • , a, -j- 1, • • • , 0,0 . More generally, reflecting a given vertex successively in symmetry planes i, j, k, ■ • ■ yields a new vertex whose coordinates differ from the original vertex precisely in places i, j, k ■ ■ ■ . Successive reflections in hyperplanes constitute a transfor- mation that leaves distances between points unaltered and is therefore a "rotation." The rotation obtained by reflecting successively in sym- metry planes ?', j, k, etc. can be represented by an ?i-place symbol having a one in places ?', j, k, etc. and a zero elsewhere. We now regard a given {n, /j)-alphabet as generated by operating on the vertex (0, 0, • • ■ , 0) of the cube with a certain collection of 2 ro- 224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 tation operators. The symbols for these operators are identical with the sequences of zeros and ones that form the coordinates of the 2 points. It is readily seen that these rotation operators form a group which is transitive on the letters of the alphabet and which leave the unit cube invariant. Theorem 2 then follows. Theorem 2 also follows readily from consideration of the array (4). For example, the maximum likelihood region associated with / is the set of points I, So , S3 , • • • , Sy . The maximum likelihood region asso- ciated with A; is the set of points Ai , AiS^ , AiSs , ■ • ■ , AiSy . The rotation (successive reflections in symmetry planes of the cube) whose symbol is the same as the coordinate sequence of Ai sends the maximum likelihood region of / into the maximum likelihood region oi Ai , i = 1, 2, • • • , M. 2.5 PROOF OF THEOREM 3 That every systematic alphabet is a group alphabet follows trivially from the fact that the sum mod 2 of two letters satisfying parity checks is again a letter satisfying the parity checks. The totality of letters satis- fying given parity checks thus constitutes a finite group. To prove that every group alphabet is a systematic code, consider the letters of a given (w, /c) -alphabet listed in a column. One obtains in this way a matrix with 2 rows and n columns whose entries are zeros and ones. Because the rows are distinct and form a group isomorphic to Ck , there are k linearly independent rows (mod 2) and no set of more than h independent rows. The rank of the matrix is therefore h. The matrix therefore possesses k linearly independent (mod 2) columns and the remaining n — k columns are linear combinations of these A;. Main- taining only these k linearly independent columns, we obtain a matrix of k columns and 2*' rows with rank k. This matrix must, therefore, have k linearly independent rows. The rows, however, form a group under mod 2 addition and hence, since k are linearly independent, all 2" rows must be distinct. The matrix contains only zeros and ones as entries; it has 2 distinct rows of k entries each. The matrix must be a listing of the num- bers from to 2^^ — 1 in binary notation. The other n — k columns of the original matrix considered are linear combinations of the columns of this matrix. This completes the proof of Theorem 3 and Proposition 4. 2.6 PROOF OF THEOREM 4 To prove Theorem 4 we first note that the parity check sequence of the product of two elements of Bn is the mod 2 sum of their separate A CLASS OF BINARY SIGNALING ALPHABETS 225 parity check sequences. It follows then that all elements in a given coset have the same parity check sequence. For, let the coset be Si , SiA2 , SiAz , ■ ■ • SiA^ . Since the elements I, A^ , A3, • • • , A^ all have parity check sequence 00 • • • 0, all elements of the coset have parity check R(Si). In the array (4) there are 2" cosets. We observe that there are 2"~* elements of Bn that have zeros in their first k places. These elements have parity check symbols identical with the last n — k places of their symbols. These elements therefore give rise to 2"~ different parity check symbols. The elements must be distributed one per coset. This proves Theorem 4. 2.7 PROOF OF PROPOSITION 5 If n ^ 2"-' - we can explicity exhibit group alphabets having the property mentioned in the paragraph preceding Proposition 5. The notation of the demon- stration is cumbersome, but the idea is relatively simple. We shall use the notation of paragraph 2.1 for elements of Bn , i.e., an element of Bn will be given by a list of integers that specify what places of the sequence for the element contain ones. It will be convenient furthermore to designate the first k places of a sequence by the integers 1, 2, 3, • • • , k and the remaining n — k places by the "integers" 1', 2', 3', • • • , r, where ( = n — k. For example, if n = 8, /c = 5, we have 10111010^ 13452' 10000100^ 11' 00000101 ^ 1'3' Consider the group generated by the elements 1', 2', 3', • • • , (' , i.e. the 2' elements /, 1', 2', ■■■,(', 1'2', 1'3', • • • , 1'2'3' ■■■('. Suppose these elements listed according to decreasing weight (say in decreasing order when regarded as numbers in the decimal system) and numbered consecutively. Let Bt be the zth element in the list. Example: if ( ^ 3, Ih = 1'2'3', B2 = 2'3', B, = 1'3', B, = 1'2', B, - 3', B, = 2', B, - 1'. Consider now the (n, /^-alphabet whose generators are ISi , 2B, ,W,, ■■• , kBk We assert that if 22G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 >r% n—k .. — 2 - this alphabet is as good as any other alphabet of 2 letters and n places. In the first place, we observe that every letter of this (n, A-)-alphabet (except /) has unprimed numbers in its symbols. It follows that each of the 2' letters /, 1', 2', • ■ • , (', V2', ■■■ , V2' ■■■ (' occurs in a different coset of the given (n, A-)-alphabet. For, if two of these letters appeared in the same coset, their product (which contains only primed numbers) would have to be a letter of the (n, k) alphabet. This is impossible since every letter of the (/i, A) alphabet has unprimed numbers in its symbol. Since there are precisely 2 cosets we can designate a coset by the single element of the list Bi , Bi , ■ • ■ , B-ii = I which appears in the coset. We next observe that the condition 71 ^ 2 — guarantees that J5a+i is of weight 3 or less. For, the given condition is equivalent to '-■-©-o-o-e We treat several cases depending on the weight of Bu+i . If Bk+\ is of weight 3, we note that for i = 1,2, • • • , A-, the coset con- taining Bi also contains an element of weight one, namely the element i obtained as the product of Bi with the letter iBi of the given (n, A;)- alphabet. Of the remaining (2 — A') 5's, one is of weight zero, C are of weight one, f j are of weight 2 and the remaining are of weight 3. We have, then an = 1, ai = f + A- = n. Now every B of weight 4 occurs in' the list of generators \Bi , 2B-2 , • • • , kBk . It follows that on multi- plying this list of generators by any B of weight 3, at least one element of weight two will result. (E.g., (l'2'3')(il'2'3'40 = j4') Thus every coset with a B of weight 2 or 3 contains an element of weight 2 and a2 = 2 — ao — cn] . The argument in case Bk+i is of weight two or one is similar. 2.8 MODULAR REPRESENTATIONS OF C„ In order to explain one of the methods used to obtain the best (//, A)- alphabets listed in Tal)les II and III, it is necessary to digress here lo present additional theory. I A CLASS OF BINARY SINGALING ALPHABETS 227 It has been remarked that every (n, /v)-alphabet is isomorphic with Ck . Let us suppose the elements of Ci, hsted in a column starting with / and proceeding in order /, 1, 2, 3, • • • , /.', 12, 13, ■••,(/.•— 1)/,-, 123, , 123 • • • k. The elements of a given (n, A-)-alphabet can be paired off with these abstract elements so as to preserve group multipli- cation. This can be done in many different ways. The result is a matrix with elements zero and one with 7i columns and 2 rows, these latter being labelled by the symbols /, 1,2, • • • etc. What can be said about the columns of this matrix? How many different columns are possible when all (n, A)-alphabets and all methods of establishing isomorphism with Ck are considered? In a given column, once the entries in rows 1,2, • • • , /,• are known, the entire column is determined by the group property. There are therefore only 2 possible different columns for such a matrix. A table showing these 2 possible columns of zeros and ones will be called a modular repre- senfafion table for Ck ■ An example of such a table is shown for /,• = 4 in Table VI. It is clear that the colunuis of a modular representation table can also be labelled by the elements of Ck , and that group multiplication of these column labels is isomorphic with mod 2 addition of the columns. The table is a symmetric matrix. The element with row label A and column label B is one if the symbols A and B have an odd number of different numerals in common and is zero otherwise. Every (n, /c)-alphabet can be made from a modular representation table by choosing w columns of the table (with possible repetitions) at least k of which form an independent set. Table VI — Modular Representation Table for Group C4 I 12 3 4 12 13 14 23 24 34 123 124 134 234 1234 I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234 n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 () 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u 228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We henceforth exclude consideration of the column / of a modular representation table. Its inckision in an (n, /v)-alphabet is clearly a waste of 1 binary digit. It is easy to show that every column of a modular representation table for Ch contains exactly 2 " ones. Since an (n, /v)-alphabet is made from n such columns the alphabet contains a total of n2 '~ ones and we have Proposition 6. The weights of an (n, /c)-alphabet form a partition of n2''~^ into 2* — 1 non-zero parts, each part being an integer from the set 1,2, ■■■ ,n. The identity element always has weight zero, of course. It is readily established that the product of two elements of even weight is again an element of even weight as is the product of two ele- ments of odd weight. The product of an element of even weight with an element of odd weight yields an element of odd weight. The elements of even weight of an (n, A;) -alphabet form a subgroup and the preceding argument shows that this subgroup must be of order 2*" or 2*""^ If the group of even elements is of order 2''~\ then the collec- tion of even elements is a possible (n, k — l)-alphabet. This (n, k — 1) alphabet may, however, contain the column / of the modular represen- tation table of Ck-i ■ We therefore have Proposition 7. The partition of Proposition 6 must be either into 2^ — 1 even parts or else into 2 " odd parts and 2^—1 even parts. In the latter case, the even parts form a partition of a2 "" where a is some integer of the set k — I, k, ■ • • , n and each of the parts is an in- teger from the set 1, 2, • • • , n. 2.9 THE CHARACTERS OF Ck Let us replace the elements of Bn (each of which is a sequence of zeros and ones) by sequences of 4-1 's and — I's by means of the following substitution The multiplicative properties of elements of Bn can be preserved iti this new notation if we define the product of two 4-1,-1 symbols to be the symbol whose tth component is the ordinary product of the ?'th compo- nents of the two factors. For example, 1011 and 01 10 become respectively -11 -1 -1 and 1 -1 -11. We have (-11 -1 -1)(1 -1 -11) = (-1 -11 -1) 1 1 -1 -1 A CLASS OF BINARY SIGNALING ALPHABETS 229 corresponding to the fact that (1011) (0110) = (1101) If the +1,-1 symbols are regarded as shorthand for diagonal matrices, so that for example -11 -1 -1 then group multiplication corresponds to matrix multiplication. (While much of what follows here can be established in an elementary way for the simple group at hand, it is convenient to fall back upon the established general theory of group representations for several proposi- tions. The substitution (13) converts a modular representation table (col- umn / included) into a square array of +l's and — I's. Each column (or row) of this array is clearly an irreducible representation of Ck ■ Since Ck is Abelian it has precisely 2 irreducible representations each of degree one. These are furnished by the converted modular table. This table also furnishes then the characters of the irreducible representations of Ck and we refer to it henceforth as a character table. Let x"(^) be the entry of the character table in the row labelled A and column labelled a. The orthogonality relationship for characters gives E x'{A)/{A) = 2'8., ACCk Z x%A)x"(B) = 2'b <xCCk AB where 8 is the usual Kronecker symbol. In particular E xiA)x\A) = Z AA) = 0, ^^I ACCk ACCk Since each x (A) is +1 or — 1, these must occur in eciual numbers in any column ^ 9^ I. This implies that each column except / of the modular representation table contains 2 ~ ones, a fact used earlier. Every matrix representation of Ck can be reduced to its irreducible components. If the trace of the matrix representing the element A in an arbitrary matrix representation of Ck is x{A), then this representation contains the irreducible representation having label ^ in the character table dp times where 230 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 (h = ^. E x{A)AA) (14) 2^- A C Ck Every (n, A)-alphabet furnishes iis with a matrix representation of Ck by means of (13) and the procedure outUned below (13). The trace xi^.) of the matrix representing the element A of C\ is related to the weight of the letter by x(A) = n - 2w(A) (15) Equations (14) and (15) permit us to compute from the weights of an (u, /,)-alphabet what irreducible representations are present in the alpha- bet and how many times each is contained. It is assumed here that the given alphabet has been made isomorphic to Ck and that the weights are labelled by elements of Ck ■ Consider the converse problem. Given a set of mmibers ivi , Wn , • ■ ■ , W'lk that satisfy Propositions 6 and 7. From these we can compute cjuantities %/ = n — 2wi as in (15). It is clear that the given ty's will constitute the weights of an (/t, A)-alphabet if and only if the 2^ x» can be labelled with elements of (\ so that the 2 sums (14) {fi ranges over all elements of Ck) are non-negative integers. The integers d^ tell what representations to choose to construct an in, A)-alphabet with the given weights Wi . 2.10 CONSTRUCTION OF BEST ALPHABETS A great many different techniques were used to construct the group alphabets listed in Tables II and III and to show that for each n and k there are no group alphabets with smaller probability of error. Space prohibits the exhibition of proofs for all the alphabets listed. We content ourseh'es here with a sample argument and treat the case n = 10, k = 4 in detail. According to (2) there are A^(10, 4) = 53,743,987 different (10, 4)- alphabets. We now show that none is better than the one given in Table III. The letters of this alphabet and weights of the letters are 1 167 8 10 5 2 6 7 9 10 5 3 5 6 8 9 10 6 4 5 7 8 9 10 6 1289 4 13579 5 A CLASS OF BINARY SIGNALING ALPHABETS 231 14569 23578 24568 3 4 6 7 12 3 5 7 9 12 4 5 7 10 1 3 4 8 10 2 3 4 9 10 12 3 4 6 7 8 9 5 5 5 4 6 6 5 5 8 The notation is that of Section 2.1. By actually forming the standard array of this alphabet, it is verified that ao =1, Oil = 10, «2 39, a:i 14. Table II shows ( .-> ) = ^5, whereas a-z = 39, so the given alphabet does not correct all possible double errors. In the standard array for the alphabet, 39 coset leaders are of weight 2. Of these 39 cosets, 33 have only one element of weight 2; the remaining 6 cosets each contain two elements of weight 2. This is due to the two elements of weight 4 in the given group, namely 1289 and 3467. A portion of the standard array that demonstrates these points is 1289 3467 12 89 • 18 29 • 19 28 . 34 67 36 47 37 46 ] • In order to have a smaller probability of error than the exhibited alphabet, it is necessary that a (10, 4)-alphabet have an a^ > 39. We proceed to show that this is impossible by consideration of the weights of the letters of possible (10, 4)-alphabets. We first show that every (10, 4)-alphabet must have at least one ele- ment (other than the identity, /) of weight less than 5. By Propositions • ') and 7, Section 2.8, the weights must form a partition of 10-8 = 80 into 1 5 positive parts. If the weights are all even, at least two must be less than 6 since 14-6 = 84 > 80. If eight of the weights are odd, we see from 8-5 + 7-() = 82 > 80 that at least one weight must be less than 5. 232 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 An alphabet with one or more elements of weight 1 must have an «2 ^ 36, for there are nine elements of weight 2 which cannot possibly be coset leaders. To see this, suppose (without loss of generality) that the alphabet contains the letter 1. The elements 12, 13, 14, • • • 1 10 can- not possibly be coset leaders since the product of any one of them with the letter 1 yields an element of weight 1 . An alphabet with one or more elements of weight 2 must have an ai S 37. Suppose for example, the alphabet contained the letter 12. Then 13 and 23 must be in the same coset, 14 and 24 must be in the same coset, ■ • • , 1 10 and 2 10 must be in the same coset. There are at least eight elements of weight two which are not coset leaders. Each element of weight 3 in the alphabet prevents three elements of weight 2 from being coset leaders. For example, if the alphabet contains 123, then 12, 13, and 23 cannot be coset leaders. We say that the three elements of weight 2 are "blocked" by the letter of weight 3. Suppose an alphabet contains at least three letters of weight three. There are several cases: (A) if three letters have no numerals in common, e.g., 123, 456, 789, then nine distinct elements of weight 2 are blocked and a-2 S 36; (B) if no two of the letters have more than a single numeral in common, e.g., 123, 345, 789, then again nine elements of weight 2 are blocked and a-2 ^ 36; and (C) if two of the letters of weight 3 have two numerals in common, e.g., 123, 234, then their product is a letter of weight 2 and l)y the preceding paragraph ao ^ 37. If an alphabet contains exactly two elements of weight 3 and no elements of weight 2, the elements of weight 3 block six elements of weight 2 and 0:2 ^ 39. The preceding argument shows that to be better than the exhibited alphabet a (10, 4)-alphabet with letters of weight 3 must have just one such letter. A similar argument (omitted here) shows that to be better than the exhibited alphabet, a (10, 4)-alphabet cannot contain more than one element of weight 4. Furthermore, it is easily seen that an alphabet containing one element of weight 3 and one element of weight 4 must have an ao ^ 39. The only new contenders for best (10, 4)-alphabet are, therefore, alphabets with a single letter other than / of weight less than 5, and this letter must have weight 3 or 4. Application of Propositions 6 and 7 show that the only possible weights for alphabets of this sort are: 35 6 and 5 46' where 5' means seven letters of weight 5, etc. We next show that there do not exist (10, 4)-alphabets having these weights. Consider first the suggested alphabet with weights 35 6'. As explained in Section 2.9, from such an alphabet we can construct a matrix repre- sentation of ('4 having the character x(/) = 10, one matrix of trace 4, A CLASS OF BINARY SIGNALING ALPHABETS 233 seven of trace and seven of trace —2. The latter seven matrices cor- respond to elements of even weight and together with / must represent a subgroup of order 8. We associate them with the subgroup generated by the elements 2, 3, and 4. We have therefore x(/) = 10, x(2) = x(3) = x(4) = x(23) = x(24) = x(34) = x(234) = -2. Examination of the symmetries involved shows that it doesn't matter how the remaining Xi ai"e associated with the remaining group elements. We take, for example x(l) = 4, x(12) = x(13) = x(14) = x(123) = x(124) = x(134) = x(1234) = 0. Now form the sum shown in equation (14) with /3 = 1234 (i.e., with the character x^" obtained from column 1234 of the Table VI by means of substitution (13). There results c?i234 = V-i which is impossible. There- fore there does not exist a (10, 4) -alphabet with weights 35 6 . The weights 5 46 correspond to a representation of d with character x(/) = 10, 0^, 2, ( — 2)^ We take the subgroup of elements of even weight to be generated by 2, 3, and 4. Except for the identity, it is clearly im- material to w^hich of these elements we assign the character 2. We make the following assignment: x(/) = 10, x(2) = 2, x(3) = x(4) = x(23) = x(24) = x(34) = x(234) = -2, x(l) = x(12) = x(13) = x(14) = x(123) = x(124) = x(134) = x(1234) = 0. The use of equation (14) shows that ^2 = \'2 which is impossible. It follows that of the 53,743,987 (10, 4)-alphabets, none is better than the one listed on Table III. Not all the entries of Table III were established in the manner just demonstrated for the (10, 4)-alphabet. In many cases the search for a l)est alphabet was narrowed down to a few alphabets by simple argu- ments. The standard arrays for the alphabets were constructed and the best alphabet chosen. For large n the labor in making such a table can be considerable and the operations involved are highly liable to error when performed by hand. I am deeply indebted to V. M. Wolontis who programmed the IBM CPC computer to determine the a's of a given alphabet and who pa- tiently ran off many such alphabets in course of the construction of Tables II and III. I am also indebted to Mrs. D. R. Fursdon who eval- uated many of the smaller alphabets by hand. 234 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 REFERENCES 1. R. W. Hamming, B.S.T.J., 29, i)p. 147-160, 1950. 2. I. S. Reed, Transactions of tlie Piofossional (iroup on Information Tlieorv, ^ PGIT-4, PI). 3S-49, 1954. 3. See section 7 of R . W. Hamniinji's paper, loc. cit. 4. I.R.E. Convention Record, I'art 4, pp. 37-45, 1955 National Convention, March, 1955. 5. C. E. Shannon, B.S.T.J., 27, pp. 379-423 and pp. 623-656, 1948. 6. Birkhoff and MacLane, A Snrvey of Modern Algebra, Macmillan Co., New York, 1941 . Van der Waerden, Alodern Algebra, Ungar Co., New York, 1953. Miller, Bliclifeldt, and Dickson, Finite Groups, Stechert, New York, 1938. 7. 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A., see Hines, M. E. Slighter, W. P.^ Proton Magnetic Resonance in Polyamides, J. Appl. Phys., 26, pp., 1099-1103, Sept., 1955. Smith, B./ and Boorse, H. A. Helium II Film Transport. II. The Role of Surface Finish, Phys. Rev. 99, pp. 346-357, July 15, 1955. Smith, B.,^ and Boorse, H. A. Helium II Film Transport. IV. The Role of Temperature, Phys. Rev., 99, pp. 367-370, July lo, 1955. SuHL, H.,^ Van Uitert, L. G.,^ and Davis, J. L.^ Ferromagnetic Resonance in Magnesium-Manganese Aluminum Fer- rite Between 160 and 1900 Mc, Letter to the Editor, J. Appl. Phys., 26, pp. 1181-1182, Sept., 1955. 1. Bell Telephone Laboratories, Inc. 6. Columbia University, New York City 240 THE EELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Thurmond, C. D., see Hassion, F. X. TiDD, W. H/ I Demonstration of Bandwidth Capabilities of Beyond -Horizon Tropo- spheric Radio Propagation, Proc. I.R.E., 43, pp. 1297-1299, Oct., 1955. Tien, P. K.,' and Walker, L. R.' Large Signal Theory of Traveling -Wave Amplifiers, Proc. I.R.E., 43, p. 1007, Aug., 1955. TiLDEN, E. F., see Bozorth, R. M. Trumbore, F. a., see Hassion, F. X. IThlir, a., Jr.^ Micromachining with Virtual Electrodes, Rev. Sci., Instr., 26, pp. 965-968, Oct., 1955. Ulrich, W., see Yokelson, B. J. Van Uitert, L. G., see Siihl, H. Walker, L. R., see Tien, P. K. Weibel, E. S.' Vowel Synthesis by Means of Resonant Circuits, J. Acous. Soc, 27, pp. 858-865, Sept., 1955. Williams, A. J., see Bozorth, R. M. WiNSLow, F. H.,' Baker, W. O.,^ and Yager, W. A.^ Odd Electrons in Polymer Molecules, Am. Chem. Soc, 77, pp. 4751- 4756, Sept. 20, 1955. WiNSLow, F. II.,' Baker, W. O.,' Rape, N. R.' and Matreyek, W.' Formation and Properties of Polymer Carbon, J. Polymer Science, 16, p. 101, Apr., 1955. Yager, W. A., sec Bozorth, R. M. 1. Bell Tc;l(;i)li()ne liaboratorics, Inc. TECHNICAL PAPERS 241 Yagkr, W. a./ Galt, J. K/ and Merritt, F. R.' Ferromagnetic Resonance in Two-Nickel-Iron Ferrites, Phys. Rev., 99, pp. 1203-1209, Aug. 15, 1955. YoKELSON, B. J.,^ and Ulrich, W.^ Engineering Multistage Diode Logic Circuits, A.I.E.E. Commun. & Electronics, 20, pp. -466-475, Sept., 1955. 1. Bell Telephone Laboratories, Inc. Recent Monographs of Bell System Technical Papers Not Published in This Journal* Arnold, W. O., and Hoefle, R. R. A System Plan for Air Traffic Control, ]\Ionograph 2483. Beck, A. C. Measurement Techniques for Multimode Waveguides, ]\Ioiiograph 2421. Becker, J. A., and Brandes, R. G. Adsorption of Oxygen on Tungsten as Revealed in Field Emission Microscope, Alonogiaph 24U3. Boyle, W. S., see Germer, L. H. Brandes, R. G., see Becker, J. A. Brattain, W. H., see Garrett, C. G. B. Garrett, C. G. B., and Brattain, W. H. Physical Theory of Semiconductor Surfaces, Monograph 2453. Gerner, L. H., Boyle, W. S., and Kisliuk, P. Discharges at Electrical Contacts — II, Monograph 2499. Hoefle, R. R., see Arnold, W. 0. KisLiuK, P., see Germer, L. H. Linvill, J. G. Nonsaturating Pulse Circuits Using Two Junction Transistors, Mono- graph 2-17."). I * Copies of these monographs may 1)0 ()l)l;tin(Ml on request to the Pul)licat ion Department, Hell Telephone Laboratories, Iiie., 463 West Street, New York 14, N. Y. The numbers of the monographs should be given in all requests. 242 MONOGRAPHS 243 Mason, W. P. Relaxations in the Attenuation of Single Crystal Lead, Monograph 2454. Mkykr, F. T. An Improved Detached-Contact-Type of Schematic Circuit Drawing, Monograph 2456. VoGEL, F. L., Jr. Dislocations in Low-Angle Boundaries in Germanium, Monograph 2455. Walker, T.. R. Generalizations of Brillouin Flow, Monograph 2432. Warner, A. W. Frequency Aging of High -Frequency Plated Crystal Units, Monograph 2474. Weibel, E. S. On Webster's Horn Equation, Monograph 2450. Contributors to This Issue A. C. Beck, E.E., Rensselaer Polytechnic Institute, 1927; Instructor, Rensselaer Polytechnic Institute, 1927-1928; Bell Telephone Labora- tories, 1928 -. With the Radio Research Department he was engaged in the development and design of short-wave and microwave antennas. During World War II he was chiefly concerned with radar antennas and associated waveguide structures and components. For several years after the war he worked on development of microwave radio repeater systems. Later he worked on microwave transmission developments for broadband communication. Recently he has concentrated on further developments in the field of broadband communication using circular waveguides and associated test equipment. J. S. Cook, B.E.E., and M.S., Ohio State University, 1952; Bell Telephone Laboratories, 1952 -. Mr. Cook is a member of the Research in High-Frequency and Electronics Department at Murray Hill and has been engaged principally in research on the traveling- wave tube. Mr. Cook is a member of the Institute of Radio Engineers and belongs to the Professional Group on Electron Devices. 0. E. DeLange, B.S. University of Utah, 1930; M.A. Columbia Uni- versity, 1937; Bell Telephone Laboratories, 1930 — . His early work was principally on the development of high-frequency transmitters and re- ceivers. Later he worked on frequency modulation and during World War II was concerned with the development of radar. Since that time he has been involved in research using broadband systems including microwa^'e and baseband. Mr. DeLange is a member of the Institute of Radio Engineers. R. KoMPFNER, Engineering Degree, Technische Hochschule, Vienna, 1933; Ph.D., Oxford, 1951; Bell Telephone Laboratories, 1951 -. Be- tween 1941-1950 he did work for the British Admiralty at Birmingham University and Oxford University in the Royal Naval Scientific Service. He invented the traveling-wave tube and for this achievement Dr. Kompfner i-eceived the 1955 Duddcll Medal, bestowed by the Physical Society of England. In the Laboratoi'ies' Research in High Frequency 244 CONTRIBUTORS TO THIS ISSUE 245 and Electronics Department, he has continued his research on vacuum tubes, particularly those used in the microwave region. He is a Fellow of the Institute of Radio Engineers and of the Physical Society in London. Charles A. Lee, B.E.E., Rensselaer Polytechnic Institute, 1943; Ph.D., Columbia University, 1953; Bell Telephone Laboratories, 1953-. When Mr. Lee joined the Laboratories he became engaged in research concerning solid state devices. In particular he has been developing techniques to extend the frequency of operation of transistors into the microwave range, including work on the diffused base transistor. During World War II, as a member of the United States Signal Corps, he was concerned with the determination and detection of enemy counter- measures in connection with the use of proximity fuses by the Allies. He is a member of the American Physical Society and the American Institute of Physics. He is also a member of Sigma Xi, Tau Beta Pi and Eta Kappa Nu. John R. Pierce, B.S., M.S. and Ph.D., California Institute of Tech- nology 1933, 1934 and 1936; Bell Telephone Laboratories, 1936-. Ap- pointed Director of Research — Electrical Communications in August, 1955. Dr. Pierce has specialized in Development of Electron Tubes and Microwave Research since joining the Laboratories. During World War li II he concentrated on the development of electronic devices for the [I Armed Forces. Since the war he has done research leading to the develop- ;j ment of the beam traveling- wave tube for which he was awarded the h 1947 Morris Liebmann Memorial Prize of the Institute of Radio Engi- [li neers. Dr. Pierce is author of two books: Theory and Design of Electron Ij Beams, published in second edition last year, and Traveling Wave Tubes il (1950). He was voted the ''Outstanding Young Electrical Engineer of [| 1942" by Eta Kappa Nu. Fellow of the American Physical Society and J the I.R.E. Member of the National Academy of Sciences, the A.I.E.E., I Tau Beta Pi, Sigma Xi, Eta Kappa Nu, the British Interplanetary So- il ciety, and the Newcomen Society of North America. C. F. QuATE, B.S., University of Utah 1944; Ph.D., Stanford Uni- i versity 1950; Bell Laboratories 1950-. Dr. Quate has been engaged in rj research on electron dynamics — the study of vacuum tubes in the ;| microwave frequency range. He is a member of I.R.E. I David Slepian, University of Michigan, 1941-1943; M.A. and Ph.D., li Harvard LTniversity, 1946-1949; Bell Telephone Laboratories, 1950-. Dr. 24G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 Slepian has been engaged in mathematical research in communication theory, switching theory and theory of noise. Parker Fellow in physics. Harvard University 1949-50. Member of I.R.E,, American Mathemati- cal Society, the American Association for the Advancement of Science and Sigma Xi. Milton Sobel, B.S., City College of New York, 1940; M.A., 1946 and Ph.D., 1951, Columbia University; U. S. Census Bureau, Statistician, 1940-41; U. S. Army War College, Statistician, 1942-44; Cohunbia Uni- versity, Department of Mathematics, Assistant, 1946-48 and Research Associate 1948-50; Wayne University, Assistant Professor of Mathe- matics, 1950-52; Columbia University, Department of Mathematical Statistics, Visiting Lecturer, 1952; Cornell University, fundamental re- search in mathematical statistics, 1952-54; Bell Telephone Laboratories, 1954-. Dr. Sobel is engaged in fundamental research on life testing reliability problems with special application to transistors and is a con- sultant on many Laboratories projects. Member of Institute of Mathe- matical Statistics, American Statistical Association and Sigma Xi. Morris Tanenbaum, A.B., Johns Hopkins University, 1949; M.A., Princeton University, 1950; Ph.D. Princeton University, 1952; Bell Telephone Laboratories, 1952-, Dr. Tanenbaum has been concerned with the chemistry and semiconducting properties of intermetallic com- pounds. At present he is exploring the semiconducting properties of silicon and the feasibility of silicon semiconductor devices. Dr. Tanen- baum is a member of the American Chemical Society and American Physical Society. He is also a member of Phi Lambda LTpsilon, Phi Beta Kappa and Sigma Xi. Donald E. Thomas, B.S. in E.E., Pennsylvania State College, 1929; M.A., Columbia University, 1932; Bell Telephone Laboratories, 1929- 1942, 1946-. His first assignment at the Laboratories was in submarine cable development. Just prior to World War II he became engaged in the development of sea and airborne radar and continued in this work I until he left for military duty in 1942. During World War II he was made ' a member of the Joint and Combined Chiefs of Staff Committees on Radio C-ountermeasures. Later he was a civilian memlior of the Depart-' ment of Defense's Research and Development Board Panel on Electronic Countermeasures. Upon rejoining the Laboratories in 1946, Mr. Thomas was active in the development and installation of the first deep sea re- peatered submarine telephone cable, hctwcen Key West and Havana,' COXTIUBUTOKS TO THIS ISSUE 247 which went into service in 1950. Later he was engaged in the develop- ment of transistor devices and circuits for special applications. At the present time he is working on the evaluation and feasibility studies of new types of semiconductors devices. He is a senior member of the I.R.E. and a member of Tau Beta Pi and Phi Kappa Phi. Laurence R. Walker, B.Sc. and Ph.D., McGill University, 1935 and 1939; LTniversity of California 1939-41; Radiation Laboratory, Massachusetts Institute of Technology, 1941-45; Bell Telephone La- boratories, 1945-. Dr. Walker has been primarily engaged in the develop- ment of microwave oscillators and amplifiers. At present he is a member of a physical research group concerned with the applied physics of solids. Fellow of the American Physical Society. IHE BELL SYSTEM Jechnical journal VOTED TO THE SC I E N T I FIC^^^ AND ENGINEERING PECTS OF ELECTRICAL COMMUNICATION LUME XXXV MARCH 1956 NUMBER 2 An Experimental Remote Controlled Line Concentrator \.f^ y A^E. JOEL, JR. 249 Transistor Circuits for Analog and Digital Systems F. H. BLECHER 295 Electrolytic Shaping of Germanium and Silicon a. uhlir, jr. 333 A Large Signal Theory of Traveling-Wave Amplifiers p. k. tibn 349 A Detailed Analysis of Beam Formation with Electron Guns of the Pierce Type w. e. danielson, j. l. rosenfeld and j. a. saloom 375 Theories for Toll Traffic Engineering in the U.S.A. r. i, Wilkinson 421 Crosstalk on Open -Wire Lines W, C, BABCOCK, ESTHER RENTROP AND C. S. THAELER 515 Bell System Technical Papers Not Published in This Journal 519 Recent Bell System Monographs 527 Contributors to This Issue 531 COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. K. K A P P E L, President Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Executive Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. MCMILLAN, Chairman A. J. BUSCH H. R. HUNTLEY A. C. DICKIBSON F. R. LACK R. L. DIETZOLD J. R. PIERCE K. E. GOULD H. V. SCHMIDT E. I. GREEN C. ESCHOOLEY R. K. HON AM AN G. N. THAYER ED ITORI AL STAFF J. D. TEBO, Editor M. E. s T R I E B Y, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Qeo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV MARCH 1956 number 2 Copyright 1958, American Telephone and Telegraph Company An Experimental Remote Controlled Line Concentrator By. A. E. JOEL, JR. (Manuscript received June 30, 1955) Concentration, which is the process of connecting a number of telephone lines to a smaller number of switching paths, has always been a funda?nental function in switching systems. By performing this function remotely from the central office, a new balance between outside plant and switching costs may be obtained which shows promise of providing service more economi- cally in some situations. The broad concept of remote line concentrators is not new. However, its solution with the new devices and techniques now available has made the possibilities of decentralization of the means for switching telephone con- nections very promising. Three models of an experimental equipment have been designed and con- structed for service. The models have included equipment to enable the evalua- tion of new procedures required by the introduction of remote line concentra- tors into the telephone plant. The paper discusses the philosophy, devices, and techniques. CONTENTS 1 . Introduction 250 2. Objectives 251 3. New Devices Emploj^ed 252 4. New Techniques Emploved 254 5. Switching Plan ". 257 249 250 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 6. Basic Circuits 261 a. Diode Gates 261 b. Transistor Bistable Circuit 262 c. Transistor Pulse Amplifier 263 d. Transistor Ring Counter 264 e. Crosspoint Operating Circuit 266 f . Crosspoint Relay Circuit 267 g. Pulse Signalling Circuit 268 h. Power Supply 269 7. Concentrator Operation 270 a.Line Scanning 270 b. Line Selection 272 c. Crosspoint Operation and Check 273 8. Central Office Circuits 274 a. Scanner Pulse Generator 279 b. Originating Call Detection and Line Number Registration 280 c. Line Selection 282 d. Trunk Selection and Identification 284 9. Field Trials 286 10. Miscellaneous Features of Trial Equipment 287 a. Traffic Recorder, b. Line Condition Tester 288 c. Simulator, d. Service Observing 290 e. Service Denial, f . Pulse Display Circuit 291 1. INTRODUCTION The equipment which provides for the switching of telephone connec- tions has ahvays been located in what have been commonly called "cen- tral offices". These offices provide a means for the accumulation of all switching equipment required to handle the telephone needs of a com- munity or a section of the community. The telephone building in which one or more central offices are located is sometimes referred to as the "wire center" because, like the spokes of a wheel, the wires which serve local telephones radiate in all directions to the telephones of the community. A new development, made possible largely by the application of de- vices and techniques new to the telephone switching field, has recently been tried out in the telephone plant and promises to change much of . the present conception of "central" offices and "wire" centers. It is known as a "line concentrator" and provides a means for reducing the amount of outside plant cables, poles, etc., serving a telephone central office by dispersing the switching equipment in the outside plant. It is not a new concept to reduce outside plant by bringing the switching equipment closer to the telephone customer but the technical difficulties of maintaining complex switching equipment and the cost of controlling" such equipment at a distance have in the past been formidable obstacles to the development of line concentrators. With the invention of low power, small-sized, long-life devices such as transistors, gas tubes, and sealed relays, and their application to line concentrators, and with the development of new local switching systems with greater flcxibilit}', it has been possible to make the progress described herein. REMOTE CONTROLLED LINE CONCENTRATOR 251 2. OBJECTIVES Within the telephone offices the first switching equipment through which dial lines originate calls concentrates the traffic to the remaining equipment which is engineered to handle the peak busy hour load with the appropriate grade of service.^ This concentration stage is different for different switching systems. In the step-by-step system^ it is the line ' finder, and in the crossbar systems it is the primary line switch.^ Pro- 1 posals for the application of remote line concentrators in the step-by- i step system date back over 50 years/ Continuing studies over the years have not indicated that any appreciable savings could be realized when such equipment is used within the local area served by a switching center. When telephone customers move from one location to another within a local service area, it is desirable to retain the same telephone numbers. The step-by-step switching system in general is a unilateral arrangement where each line has two appearances in the switching equipment, one for originating call concentration (the line finder) and one for selection of the line on terminating calls (the connector) . The connector fixes the line number and telephone numbers cannot be readily reassigned when moving these switching stages to out-of-office locations. Common-control systems^ have been designed with flexibility so that the line number assignments on the switching equipment are independ- ent of the telephone numbers. Furthermore, the first switching stage in the office is bilateral, handling both originating and terminating calls through the same facilities. The most recent common-control switching system in use in the Bell System, the No. 5 crossbar,^ has the further advantage of universal control circuitry for handling originating and terminating calls through the line switches. For these reasons, the No. 5 crossbar system was chosen for the first attempt to employ new tech- niques of achieving an economical remote line concentrator. A number of assumptions were made in setting the design require- ments. Some of these are influenced by the characteristics of the No. 5 crossbar system. These assumptions are as follows: 1. No change in customer station apparatus. Standard dial telephones to be used with present impedance levels, transmission characteristics, dial pulsing, party identification, superimposed ac-dc ringing,^ and sig- naling and talking ranges. 2. Individual and two-party (full or semi-selective ringing) stations to be served but not coin or PBX lines. 3. Low cost could best be obtained by minimizing the per line equipment in the central office. AMA^ charging facilities could be used but to avoid per station equipment in the central office no message reg- ister operation would be provided. 252 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 4. Each concentrator would serve up to 50 lines with the central office control circuits common to a number of concentrators. (Experimental equipment described herein was designed for 60 lines to provide addi- tional facilities for field trial purposes.) No extensive change would be made in central office equipment not associated with the line switches nor should concentrator design decrease call carrying capacities of exist- ing central office equipment. 5. To provide data to evaluate service performance, automatic traffic recording facilities to be integrated with the design. 6. Remote equipment designed for pole or wall mounting as an addi- tion to existing outside plant. Therefore, terminal distribution facilities would not be provided in the same cabinet. 7. Power to be supplied from the central office to insure continuity of telephone service in the event of a local power failure. 8. Concentrators to operate over existing types of exchange area fa- cilities without change and with no decrease in station to central office service range. 9. Maintenance effort to be facilitated by plug-in unit design using the most reliable devices obtainable. 3. NEW DEVICES EMPLOYED »! I Numerous products of research and development were available for this new approach. Only those chosen will be described. For the switching or "crosspoint" element itself, the sealed reed switch was chosen, primarily because of its imperviousness to dirt.* A short coil magnet with magnetic shield for increasing sensitivity of the reed switches were used to form a relay per crosspoint (see Fig. 1). A number of switching applications^ '^^ for crosspoint control using small gas diodes have been proposed by E. Bruce of our Switching Re- search Department. They are particularly advantageous when used in an "end marking" arrangement with reed relay crosspoints. Also, these diodes have long life and are low in cost. One gas diode is employed for operating each crosspoint (see Fig. 6). Its breakdown voltage is 125v ± lOv, A different tube is used in the concentrator for detecting marking potentials when termination occurs. Its breakdown potential is lOOv ± lOv. One of these tubes is used on each connection. Signaling between the remote concentrator and the central office con- trol circuits is performed on a sequential basis with pulses indicative of the various line conditions being transmitted at a 500 cycle rate. This frequency encounters relatively low attenuation on existing exchange area wire facilities and j^et is high enough to transmit and receive in- formation at a rate which will not decrease call carrjdng capacitj^ of the REMOTE CONTROLLED LINE CONCENTRATOR 253 Fig. 1 — Reed switch relay. central office equipment. To accomplish this signaling and to process the information economically transistors appear most promising. Germanium alloy junction transistors were chosen because of their ; improved characteristics, reliability, low power requirements, and mar- gins, particularly when used to operate with relays.^^ Both N-P-N and P-N-P transistors are used. High temperature characteristics are par- ticularly important because of the ambient conditions which obtain on pole mounted equipment. As the trials of this equipment have progressed, 254 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table I— Transistor Characteristics Code No. Type and Filling Alpha Max. Ico at 28V and 65°C Emitter Zener Voltage at 20^=1 M1868 M1887 p-n-p Oxygen n-p-n Vacuum 0.9-1.0 0.5- .75 150 Ma 100 Ma >735 >735 considerable progress has been made in improving transistors of thi.s type. Table I summarizes the characteristics of these transistors. For directing and analyzing the pulses, the control employs semicon- ductor diode gate circuits." The semiconductor diodes used in these circuits are of the silicon alloy junction type,^^ Except for a few diode.s operating in the gas tube circuits most diodes have a breakdown voltage requirement of 27v, a minimum forward current of 15 ma at 2v and a maximum reverse current at 22v of 2 X 10^^ amp. 4. new techniques employed The concentrator represents the first field application in Bell System telephone switching systems which departs from current practices and techniques. These include: Fig. 2 — Transistor packages, (a) Diode unit, (b) Transistor counter, (c) Transistor amplifiers and bi-stable circuits, (d) Five trunk unit. REMOTE CONTROLLED LINE CONCENTRATOR 255 1. High speed pulsing (500 pulses per second) of information between switching units. 2. The use of plug-in packages employing printed wiring and encap- sulation. (Fig. 2 shows a representative group of these units.) 3. Line scanning for supervision with a passive line circuit. In present systems each line is equipped with a relay circuit for detecting call orig- inations (service requests) and another relay (or switch magnet) for indicating the busy or idle condition of the line, as shown in Fig. 3(a). The line concentrator utilizes a circuit consisting of resistors and semi- conductor diodes in pulse gates to provide these same indications. This circuit is shown in Fig. 3(b). Its operation is described later. The pulses for each line appear at a different time with respect to one another. These pulses are said to represent "time slots." Thus a different line is examined each .002 second for a total cycle time (for 60 lines) of .120 second. This process is known as "line scanning" and the portion of the circuit which produces these pulses is known as the scanner. Each of the circuits perform the same functions, viz., to indicate to the central office equipment when the customer originates a call and for terminating calls to indicate if the line is busy. 4. The lines are divided for control and identification purposes into twelve groups of five lines each. Each group of five lines has a different pattern of access to the trunks which connect to the central office. The ten trunks to the central office are divided into two groups as shown in Fig. 4. One trunk group, called the random access group, is arranged in a random multiple fashion, so that each of these trunks is available to approximately one-half of the lines. The other group, consisting of two trunks, is available to all lines and is therefore called the full access group. The control circuitry is arranged to first select a trunk of the random access group which is idle and available to the particular line to which a connection is to be made. If all of the trunks of this random ac- cess group are busy to a line to which a connection is desired, an attempt is then made to select a trunk of the full access group. The preference order for selecting cross-points in the random access group is different for each line group, as shown in the table on Fig. 4. By this means, each trunk serves a number of lines on a different priority basis. Random ac- cess is used to reduce by 40 per cent the number of individual reed relay crosspoints which would otherwise be needed to maintain the quality of service desired, as indicated by a theory presented some years ago.^^ 5. Built-in magnetic tape means for recording usage data and making call delay measurements. The gathering of this data is greatly facilitated by the line scanning technique. 256 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CROSSBAR CROSSPOINT OR SWITCH CONTACTS -^ TO LINE -^ TO OTHER CENTRAL OFFICE EQUIPMENT 9 9 r ^ LR ■^f- CO c HI "H 1_ (a) ■:l LINE BUSY SERVICE REQUEST I + 5V CROSSPOINT ■^ TO LINE -^ TO CENTRAL OFFICE ^4- -X -16V -16 VOLTS -NORMAL (RECEIVER ON HOOK) -3 VOLTS -AWAITING SERVICE (RECEIVER OFF HOOK) -16 VOLTS -CROSSPOINT CLOSED (RECEIVER OFF HOOK) S\. -¥^ -16 V ^ LINE BUSY + 15 VOLT TIME SLOT PULSE FROM SCANNER GATE SERVICE REQUEST Fig. 3 — (a) Relay line circuit, (b) Passive line circuit. REMOTE CONTROLLED LINE CONCENTRATOR 257 5. SWITCHING PLAN The plan for serving lines directly terminating in a No. 5 Crossbar office is shown in Fig. 5(a). Each line has access through a primary line switch to 10 line links. The line links couple the primary and secondary switches together so that each line has access to all of the 100 junctors to the trunk link switching stage. Each primary line switch group accommodates from 19 to 59 lines (one line terminal being reserved for no-test calls). A line link frame contains 10 groups of primary line switches.^* . The remote concentrator plan merely extends these line links as trunks to the remote location. However, an extra crossbar switching stage is introduced in the central office to connect the links to the secondary line switches with the concentrator trunks as shown in Fig. 5(b). Since each line does not have full access to the trunks, the path chosen by the marker to complete calls through the trunk link frame may then be independent of the selection of a concentrator trunk with access to the line. This arrangement minimizes call blocking, simplifies the selection of a matched path by the marker, and the additional crossbar switch hold magnet serves also as a supervisory relay to initiate the transmission of disconnect signals over the trunk. In addition to the 10 concentrator trunks used for talking paths, 2 additional cable pairs are provided from each concentrator to the central office for signaling and power supply purposes. The use of these two pairs of control conductors is described in detail in Section 6g. The concentrator acts as a slave unit under complete control of the central office. The line busy and service request signals originate at the LINE 60 LINES I o.-»-o 5 9 7 '^ / ^ / ■v / p, ■^ i' \. ^ V •y \ / s f \ ^ \ ,• \ '^ ^ 1 > f < > 1 2 3 5 6 8 9 10 11 Fig 4. — Concentrator trunk to line crosspoint pattern and preference order CONCENTRATOR TRUNKS 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 6 5 4 7 5 3 1 4 7 2 1 7 3 1 5 2 6 4 6 5 3 1 7 2 3 6 2 4 6 3 5 4 6 2 3 7 1 6 2 4 1 7 1 5 6 8 VERTICAL GROUPS OF FIVE LINES EACH " ORDER OF PREFERENCE GAS TUBE REED -RELAY CROSS POINTS 10 11 258 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 49 LINES TEN GROUPS OF LINES 49 LINES I CENTRAL OFFICE LINE LINK FRAME LINE SW 1 CON- NECTOR 1 1 1 — -- 1 1 CON- NECTOR I TO MARKER TRUNK LINK FRAME Fig. 5(a) — No. 5 crossbar system subscriber lines connected to line link frame. 60 LINES 60 LINES 60 <? 10 CONTROL CE^4TRAL OFFICE TEN CONCENTRATOR , TRUNKS I JL TWO CONTROL PAIRS 60 10 TEN POLE- MOUNTED ^CONCENTRATOR UNITS AT DIFFERENT LOCATIONS CONTROL TEN CONCENTRATOR TRUNKS TWO CONTROL PAIRS CONCENTRATOR TRUNK SW JUNCTOR SW 10 9 C> TRUNK LINK FRAME TO MARKER CONCENTRATOR LINE LINK FRAME Fig. 5(b) — No. 5 crossbar system subscriber lines connected to remote line concentrators. REMOTE CONTROLLED LINE CONCENTRATOR 259 Fig. 6 — Line unit construction. concentrator only in response to a pulse in the associated time slot or when a crosspoint operates (a line busy pulse is generated under this condition as a crosspoint closure check). The control circuit in the central office is designed to serve 10 remote line concentrators connected to a single line link frame. In this way the marker deals with a concen- trator line link frame as it would with a regular line link frame and the marker modifications are minimized. The traffic loading of the concentrator is accomplished by fixing the 260 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Fig. 7(a) — Line unit. number of trunks at 10 and equipping or reassigning lines as needed to obtain the trunk loading for the desired grade of service. The six cross- points, the passive line circuit and scanner gates individual to each line are packaged in one plug-in unit to facilitate administration. The cross- points are placed on a printed wiring board together with a comb of plug contacts as shown in Fig. 6. The entire unit is then dipped in rubber and encapsulated in epoxy resin, as shown in Fig. 7(a). This portion of the unit is extremely reliable and therefore it may be considered as expendable, should a rare case of trouble occur. The passive line circuit and scanner gate circuit elements are mounted on a smaller second printed wiring plate (known as the "line scanner" plate, see Fig. 7(b) which fits into a recess in the top of the encapsulated line unit. Cir- Fig. 7(b) — Scanner plate of the line unit shown in Fig. 7 (a). REMOTE CONTROLLED LINE CONCENTRATOR 261 cuit connection between printed wiring plates is through pins which ap- pear in the recess and to which the smaller plate is soldered. 6. BASIC CIRCUITS a. Diode Gates All high speed signaling is on a pulse basis. Each pulse is positive and approximately 15 volts in amplitude. There is one basic type of diode gate circuit used in this equipment. By using the two resistors, one con- denser and one silicon alloy junction diode in the gate configuration shown in Fig. 8, the equivalents of opened or closed contacts in relay circuits are obtained. These configurations are known respectively as enabling and inhibiting gates and are shown with their relay equivalents ill Figs. 8(a) and 8(b). In the enabling gate the diode is normally back biased by more than the pulse voltage. Therefore pulses are not transmitted. To enable or INPUT ENABLING GATE CIRCUIT CI OUTPUT (a) ENABLING GATE SYMBOL INPUT OUTPUT CONTROL EQUIVALENT RELAY CIRCUIT OUTP UT INPUT f CONTROL CHli^ INPUT INHIBITING GATE CIRCUIT Cl OUTPUT INHIBITING GATE SYMBOL INPUT OUTPUT CONTROL EQUIVALENT RELAY CIRCUIT OUTPUT DhUHl Fig. 8 — Gates and relay equivalents. 262 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 open the gate the back bias is reduced to a small reverse voltage which is more than overcome by the signal pulse amplitude of the pulse. The pulse thus forward biases the diode and is transmitted to the output. The inhibiting gate has its diode normally in the conducting state so that a pulse is readily transmitted from input to output. When the bias is changed the diode is heavily back biased so that the pulse amplitude is insufficient to overcome this bias. The elements of 12 gates are mounted on a single printed wiring board w4th plug-in terminals and a metal enclosure as shown in Fig. 2(a). All elements are mounted in one side of the board so that the opposite side may be solder dipped. After soldering the entire unit (except the plug) is dipped in a silicone varnish for moisture protection. b. Transistor Bistable Circuit Transistors are inherently well adapted to switching circuits using but two states, on (saturated) or off.^^ In these circuits with a current gain greater than unity a negative resistance collector characteristic can be obtained which will enable the transistor to remain locked in its conduct- ing state (high collector current flowing) until turned off (no collector current) by an unlocking pulse. At the time the concentrator develop- ment started only point contact transistors were available in quantity. Point contact transistors have inherently high current gains (>1) but the collector current flowing when in the normal or unlocked condition (Ico) was so great that at high ambient temperatures a relay once op- erated in the collector circuit would not release. Junction transistors are capable of a much greater ratio of on to off current in the collector circuit. Furthermore their characteristics are amenable to theoretical design consideration.^^ However, the alpha of a simple junction transitor is less than unity. To utilize them as one would | a point contact transitor in a negative resistance switching circuit, a combination of n-p-n and p-n-p junction transistors may be employed, i see Fig. 9(b). Two transistors combined in this manner constitute a ' "hooked junction conjugate pairs." This form of bi-stable circuit was j used because it requires fewer components and uses less power than an Eccles-Jordan bistable circuit arrangement. It has the disadvantage of a single output but this was not found to be a shortcoming in the design of circuits employing pulse gates of the type described. In what follows the electrodes of the transistor will be considered as their equivalents shown in Fig. 9(b). The basic bi-stable circuit employed is shown in Fig. 10. The set REMOTE CONTROLLED LINE CONCENTRATOR 263 EMITTER COLLECTOR EMITTER n-p-n COLLECTOR BASE fa) POINT CONTACT TRANSISTOR Ic BASE (b) CONJUGATE PAIR ALLOY JUNCTION TRANSISTORS C _ 0C> 1 Fig. 9 — Point contact versus hooked conjugate pair. pulse is fed into the emitter (of the pair) causing the emitter diode to conduct. The base potential is increased thus increasing the current flowing in the collector circuit. When the input pulse is turned off the base is left at about —2 volts thus maintaining the emitter diode con- ( lucting and continuing the increased current flow in the collector circuit. The diode in the collector circuit prevents the collector from going positive and thereby limits the current in the collector circuit. To reset, a positive pulse is fed into the base through a pulse gate. The driving of tlie base positive returns the transistor pair to the off condition. c. Transistor Pulse Amplifier This circuit (Fig. 11) is formed by making a bi-stable self resetting circuit. It is used to produce a pulse of fixed duration in response to a TRANSISTORS p-n-p SET RESET I-5V -I6V F/F Fig. 10 — Transistor bi-stable circuit. 264 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 pulse of variable width (within limits) on the input. Normally the emitter is held slightly negative with respect to the base. The potential difference determines the sensitivity of the amplifier. When a positive input pulse is received, the emitter diode conducts causing an increase in collector current. The change in bias of the diode in the emitter circuit permits it to conduct and charge the condenser. With the removal of the input pulse the discharge of the condenser holds the transistor pair on. The time constant of the circuit determines the on time. When the emitter potential falls below the base potential, the transistor pair is turned off. The amplifiers and bi-stable circuits or flip-flops, >as they are called more frequently, are mounted together in plug-in packages. Each pack- age contains 8 basic circuits divided 7-1, 6-2, or 2-6, between amplifiers and fhp-flops. Fig. 2(c) shows one of these packages. They are smaller than the gate or line unit packages, having only 28 terminals instead of 42. The transistors for the field trial model w^ere plugged into small hear- ing aid sockets mounted on the printed wiring boards. For a production model it w^ould be expected that the transistors w^ould be soldered in. d. Transistor Ring Counter By combining bi-stable transistor and diode pulse gate circuits to- gether in the manner shown in Fig. 12 a ring counter may be made, with INPUT p-n-p ^w ^vW-" I + 5V OUTPUT -16 V INPUT OUTPUT Fig. 11 — Transistor pulse amplifier. REMOTE CONTROLLED LINE CONCENTRATOR 265 COUNT INPUT lie STAGE NUMBER 3 NOTE: LEADS A-0 TO A-4 ARE OUTPUT LEADS OF RESPECTIVE STAGES 1 I I \ r s 's 's 's 's Fig. 12 — Ring counter schematic. a bi-stable circuit per stage. The enabling gate for a stage is controlled by the preceding stage allowing it to be set by an input advance pulse. The output signal from a stage is fed back to the preceding stage to turn it off. An additional diode is connected to the base of each stage for re- setting when returning the counter to a fixed reference stage. A basic package of 5 ring counter stages is made up in the same frame- work and with the same size plug as the flip-flop and amplifier packages, see Fig. 2(b). A four stage ring counter is also used and is the same package with the components for one stage omitted. The input and out- put terminals of all stages are available on the plug terminals so that the stages may be connected in any combination and form rings of more than 5 stages. The reset lead is connected to all but the one stage which is considered the first or normal stage. Other transistor circuits such as binary counters and square wave generators are used in small quantity in the central office equipment. They will not be described. 266 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CONCENTRATOR LINE BUSY CENTRAL OFFICE TO ALL CROSSPOINTS / SERVED BY TRUNK + 130 V VG VF I L..1.. / TO ALL CROSSPOINTS FOR SAME LINE SELECTION FROM " CENTRAL OFFICE i-65V I + 100V Fig. 13 — Crosspoint operating circuit. e. Crosspoint Operating Circuit The crosspoint consists of a reed relay with 4 reed switches and a gas diode (Fig. 1). The selection of a crosspoint is accomplished by marking with a negative potential ( — 65 volts) all crosspoints associated with a line, and marking with a positive potential ( + 100 volts) all crosspoints associated with a trunk (Fig. 13). The line is marked through a relay circuit set by signals sent over the control pair from the central office. The trunk is marked b}^ a simplex circuit connected through the break contacts of the hold magnet of the crossbar switch associated with the trunk in the central office. Only one crosspoint at a time is exposed to 165 volts which is necessary and sufficient to break down the gas diode to its conducting state. The reed relay operates in series with the gas diode. A contact on the relay shunts out the gas diode. When the marking- potentials are removed the relay remains energized in a local 30-voll circuit at the concentrator. The holding current is approximately 2.5 ma. This circuit is designed so that ringing signals in the presence or ab- sence of lino marks will not falsely fire a crosspoint diode. Furthonnoi'o, REMOTE CONTROLLED LINE CONCENTRATOR 267 a line or trunk mark alone should not be able to fire a crosspoint diode on a busy line or trunk. When the crosspoint operates, a gate which has been inhibiting pulses is forward biased by the —65 volt signal through the crosspoint relay winding. The pulse which initiates the mark operations at the concentra- tor then passes through the gate to return a line busy signal to the central office over this control pairs which is interpreted as a crosspoint closure check signal. f. Crosspoint Release Circuit The hold magnet of the central office crossbar switch operates, remov- ing the +100- volt operate mark signal after the crosspoint check signal is received. A slow release relay per trunk is operated directly by the hold magnet. When the central office connection in the No. 5 crossbar system releases, the hold magnet is released. As shown in Fig. 14, with the hold magnet released and the slow release relay still operated, a — 130- volt signal is applied in a simplex circuit to the trunk to break down a gas tube provided in the trunk circuit at the concentrator. This tube in CONCENTRATOR CENTRAL OFFICE TO ALL CROSSPOINTS SERVED BY SAME TRUNK 130V I Fig. 14 — Crosspoint release circuit. 268 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 breaking down shunts the local holding circuit of the crosspoint causing it to release. The — 130-volt disconnect signal is applied during the release time of the slow release relay which is long enough to insure the release of the crosspoint relay at the concentrator. The release circuit is individual to the trunk and independent of the signal sent over the control pairs. g. Pulse Signalling Circuits To control the concentrator four distinct pulse signals are transmitted from the central office. Two of these at times must be transmitted simultaneously, bvit these and the other two are transmitted mutually exclusively. In addition, service request and line busy signals are trans- mitted from the concentrator to the central office. The two way trans- mission of information is accomplished on each pair by sending signals in each direction at different times and inhibiting the receipt of signals when others are being transmitted. To transmit four signals over two such pairs, both positive and nega- CONTROL PAIR NO. 1 VF M LB D SR -16V VG CONTROL PAIR NO. 2 16 V M CONCENTRATOR AMPLIFIERS I CENTRAL OFFICE AMPLIFIERS PER CONCENTRATOR Fig. 15. — Signal transmission circuit. REMOTE CONTROLLED LINE CONCENTRATOR 269 tive pulses are employed. Diodes are placed in the legs of a center tapped transformer, as shown in Fig. 15, to select the polarity of the trans- mitted pulses. At the receiving end the desired polarity is detected by taking the signal as a positive pulse from a properly poled winding of a transformer. The amplifier, as described in Section 6c responds only to positive pulses. If pulses of the same polarity are transmitted in the other direction over the same pair, as for control pair No. 1, the outputs of the receiving amplifier for the same polarity pulse are inhibited whenever a pulse is transmitted. As shown in Fig. 15, the service request and line busy signals are transmitted from the concentrator to the central office over one pair of conductors as positive and negative pulses respectivel3^ The trans- mission of these pulses gates the outputs of two of the receiving ampli- fiers at the concentrator to permit the receipt of the polarized signals from the central office. This prevents the pulses from being used at the sending end. A similar gating arrangement is used with respect to the signals when sent over this control pair from the central office. The pulses designated VG or RS never occur when a pulse designated SR or LB is sent in the opposite direction. The transmission of the VF pulse over control pair No. 2 is processed by the concentrator circuit and becomes the SR or LB pulses. Li section 7 the purpose of these pulses is described. The signaling range objective is 1,200 ohms over regular exchange area cable including loaded facilities from sfation to central office. h. Power Supply Alternating current is supplied to the concentrator from a continuous service bus in the central office. The power supply path is a phantom circuit on the two control pairs as shown in Fig. 16. The power trans- former has four secondary windings used for deriving from bridge rectifiers four basic dc voltages. These voltages and their uses are as fofiows: —16 volts (regulated) for transistor collector circuits and gate biases, -|-5 volts (regulated) for transistor base biases, -|-30 volts (regu- , lated) for crosspoints holding circuits and — 65 volts for the marking and operating of the line crosspoints. For this latter function a reference to the central office applied -flOO volt trunk mark is necessary. The refer- ence ground for the concentrator is derived from ground applied to a simplex circuit on the power supply phantom circuit. Series transistors and shunt silicon diodes with fixed reference breakdown voltages are I used to regulate dc voltages. 270 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Total power consumption of the concentrator is between 5 and 8 watts depending upon the number of connections being held. 7. CONCENTRATOR OPERATION a. Line Scanning The sixty lines are divided into 12 groups of 5 lines each. These group- ings are designated VG and VF respectively corresponding to the vertical group and file designations used in the No. 5 crossbar system. Each concentrator corresponds to a horizontal group in that system. To scan the lines two transistor ring counters, one of 12 stages and one of 5 stages, are employed as shown in Fig. 17. These counters are driven from pulses supplied from the central office control circuits and only one stage in each is on at any one time. The steps and combinations of these counters correspond to the group and file designation of a par- ticular line. Each 0.002 second the five stage counter (VF) takes a step and between the fifth and sixth pulse the r2-stage counter (VG) is stepped. Thus the 5-stage counter receives 60 pulses or re-cycles 12 times in 120 milliseconds while the 12-stage counter cycles but once. Each line is provided with a scanner gate. The collector output of each each stage of the VG counter biases this gate to enable pulses which are generated by the collector circuit of the 5-stage counter to pass on -65V + 30V + SV -I6V 115 V AC MOTOR GENERATOR TO COMMERCIAL AC REGULATORS Fig. 16 — Power supply transmission circuit. REMOTE CONTROLLED LINE CONCENTRATOR 271 to the gate of the passive line circuit, Fig. 3(b). If the line is idle the pulses are inhibited. If the receiver is off-hook requesting service (no (•rosspoint closed) then the gate is enabled, the pulse passes to the service request amplifier and back to the central office in the same time slot as the pulse which stepped the VF counter. If the line has a receiver off-hook and is connected to a trunk the pulse passes through a contact of the crosspoint relay to the line busy amplifier and then to the central office in the same time slot. At the end of each complete cycle a reset pulse is sent from the central office. This pulse instead of the VG pulse places the 12-stage counter in its first position. It also repulses the 5 stage VF counter to its fifth stage so that the next VF pulse will turn on its first stage to start the next j cycle. The reset pulse insures that, in event of a lost pulse or defect in a counter stage, the concentrator will attempt to give continuous ser- \'ice without dependence on maintaining synchronism with the central I office scanner pulse generator. Fig. 18(a) shows the normal sequence of I line scanning pulses. , When a service request pulse is generated, the central office circuits t] 04 r VF 5- STAGE COUNTER 03 TO 10 INTERMEDIATE GATES EACH V 02 I 01 00 TO 5 GATES EACH I 1 23456 789 10 I I I I I I I I I I I I I I I I I I I I VG 12-STAGE COUNTER 59\ 58 57 56 55 GATE PER LINE ■ FEEDS PASSIVE LINE CIRCUITS / VG RES ET VF FROM CENTRAL OFFICE Fig. 17 — Diode matrix for scanning lines. 272 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 common to 10 concentrators interrupt the further transmission of the vertical group pulse so that the line scanning is confined to the 5 lines in the vertical group in which the call originated. In this way the cen- tral office will receive a service request pulse at least every 0.010 sec as a check that the call has not been abandoned while awaiting service. Fig. 18(b) shows the detection of a call origination and the several short scan cycles for abandoned call detection. b. Line Selection When the central office is ready to establish a connection at the con- centrator a reset pulse is sent to return the counters to normal. In gen- eral, the vertical group and vertical file pulses are sent simultaneously to reduce holding time of the central office equipment and to minimize marker delays caused by this operation. For this reason the VG and VF pulses are each transmitted over different control pairs from the central office. The same polarity is used. On originating calls it is desirable to make one last check that the call has not been abandoned, while on terminating calls it is necessary L* 120MS >| ■M k-2MS I PULSE ' 012340123401234 0123401234 VF - VG LB RS J__l I I I I I I I I I I I I I I 1 I I I I 1 I L 1^ \1 ^ (a) REGULAR LINE SCANNING VF 123401 23401 23401 2340123401 _l_l 1 I I I I I I I I I I I I I I I I I L_l I I I I ,5 ,6 VG 1 1 — LB- RS- SR- M- (b) CALL ORIGINATION SERVICE REQUEST FROM LINE 6/3 12 3 1 I I I I I VG- LB RS — 1>- SR h 1° 1' 1^ |3 1^ 1^ 1^ jLi. M - H^-l^-- "7 RESULTS FROM CONC CONTROL RECEIVED 'OPERATE ""CROSSPOINT 'NORMAL SCANNING CKT AT CENTRAL OFFICE ONLY IF CROSSPOINT CLOSURE IS RESUMED RECEIVING FROM MKR VG , LINE 6/3 HAS INDICATION VF, HG INFORMATION BECOME BUSY (C) LINE SELECTION FOR LINE 6/3 Fig. 18 — Pulse sequences, (a) Regular, (b) Call origination, (c) Line selection. REMOTE CONTROLLED LINE CONCENTRATOR 273 to determine if the line is busy or idle. These conditions are determined in the same manner as described for line scanning since a service re- quest condition would still prevail on the line if the call was not aban- doned. If the line was busy, a line busy condition would be detected. However to detect these conditions a VF pulse must be the last pulse transmitted since the stepping of the VF counter generates the pulse which is transmitted through an enabled line selection and passive line circuit gates. Fig. 18(c) shows a typical line selection where the num- ber of VF pulses is equal to or less than the number of VG pulses. In all other cases there is no conflict and the sending of the last VF pulse need not be delayed. On terminating calls, the line busy indication is returned to the central office within 0.002 sec after the selection is com- plete. During selections the central office circuits are gated to ignore any extraneous service request or line busy pulses produced as a result of steps of the VF counter prior to its last step. c. Crosspoint Operation and Check Associated with each concentrator transistor counter stage is a reed relay. These relays are connected to the transistor collector circuits through diodes of the counter stages when relay M operates. The con- tacts of these reed relays are arranged in a selection circuit as shown in Fig. 19 and apply the —65 volt mark potential to the crosspoint relays of the selected line. After a selection is made as described above a "mark" pulse is sent from the central office. This pulse is transmitted as a pulse of a different polarity over the same control pair as the VF pulses. The received pulse after amplification actuates a transistor bistable circuit w^hich has the M reed relay permanently connected in its collector circuit. The bi-stable circuit holds the M relay operated during the crosspoint opera- tion to maintain one VF and one VG relay operated, thereby applying — 65 volts to mark and operate one of the 6 crosspoint relays of the selected line as described in section 6e, and shown on Fig. 13. The operation and locking of the crosspoint relay with the marking potentials still applied enables a pulse gate associated with the holding circuit of the crosspoint relays in each trunk circuit. The mark pulses are sent out continuously. This does not affect the bi-stable transistor circuit once it has triggered but the mark pulse is transmitted through the enabled crosspoint closure check gate shown in Fig. 20 and back to the central office as a line busy signal. With the receipt of the crosspoint closure check signal the sending 274 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of the mark pulses is stopped and a reset pulse is sent to the concentra- tor to return the mark bi-stable circuit, counters and all operated selec- tor relays to normal. The concentrator remains in this condition until it is resynchronized with the regular line scanning cycle. A complete functional schematic of the concentrator integrating the circuits described above is shown in Fig. 21. Fig. 22(a) and (b) show an experimental concentrator built for field tests. 8. CENTRAL OFFICE CIRCUITS The central office circuits for controling one or more concentrators are composed of wire spring relays as well as transistors, diode and reed VG RS VF M -20V -20 V o- VF-5 STAGE COUNTER r -65V n 04 03 02 00 6 RELAY I W-, wo w<- ,^, -o p PACKAGE J '-|„p_„p_^_p-_-p-^Ui TO CONTACTS OF 4 INTERMEDIATE RELAYS 6 RELAY PACKAGE TO CONTACTS OF 4 . INTERMEDIATE RELAYS n L 59 n 58 i^ 34 33 32 31 30 , 29 28 27 26 25 I 5^ 57 56 55 TO 4 INTERMEDIATE RELAYS ' -/ *■ I I I 2 b^'" / 7_ 8_ 9 10 cr LU h- z o o < I- OJ I (J > I TO 4 INTERMEDIATE il RELAYS P^ig. 19 — Line selection and marking. I REMOTE CONTROLLED LINE CONCENTRATOR 275 relay packages similar to those used in the concentrator. The reed relays are energized by transistor bi-stable circuits in the same manner as described in Section 7c. The reed relay contacts in turn operate wire spring relays or send the dc signals directly to the regular No. 5 crossbar marker and line link marker connector circuits. Fig. 23 shows a block diagram of the central office circuits. A small amount of circuitry is provided for each concentrator. It consists of the following: 1. The trunk connecting crossbar switch and associated slow relays for disconnect control. 2. The concentrator control triuik circuits and associated pulse ampli- fiers. 3. An originating call detector to identify which concentrator among the ten served by the frame is calling. 4. A multicontact relay to connect the circuits individual to each concentrator with the common control circuits associated with the line link frame and markers. The circuits associated with more than one concentrator are blocked out in the lower portion of Fig. 23. Much of this circuitry is similar to the relay circuits now provided on regular line link frames in the No. 5 crossbar system.^ Only those portions of these blocks which employ the new techniques will be covered in more detail. These portions consist of the following: 1. The scanner pulse generator. 2. The originating line number register. T TO ALL TRUNK LINES + 30V A/vV U j^Wv- -65V T I I I I I CONCENTRATOR TRUNK I I I I I i Fig. 20 — Crosspoint closure check. aoidzio nvbiNBo oi iinoaio ONnvNois via 276 Fig. 22(a) — Complete line concentrator unit. r 5 -STAGE COUNTER 12 -STAGE COUNTER -fO TRUNK CiftCUITS AMPLIFIERS RECTIFIERS Fig. 22(b) — Identification of units within the line concentrator. 277 a. Q-tU O z o o 1 2 o< z o CO UJ O <:^ o: u tu z D. o (J UJ£t Q. 10 I ^-; C-: n W liJO IOq Q I I I J o o LJJ o u a. o ^ cc I- z UJ o z o u Z 111 Di- ce I- l-< >^av/^l Asng i3S3a 9A dA VAVVV tr zy= Di- CEZ I-LIJ Nl 9H lAIS "11 Z CEUJ I 1 UJ CO S3 (O u 90 ui- UJ z -lO ujo (/I Hj UJ(- ZU -iUJ 9A HH dA d: p uJuJt; z -id: < DUJ en "J ; < UJ CC O O UJ zmCE ^UjS cc 2 cc On IdA cc o I- O UJ z z O O < CC U. z _J UJ ^^ 1- O u o _l UJUJ ^tr oruj <u. ^Si Q. tn IT UJ :£. .CC < 2 o I- 3A IS i9H F-^: I I a0J.VdlN33N00 Oi §(0 u cc o I- ?i UJ O zO -Jq: CC < 2 278 REMOTE CONTROLLED LINE CONCENTRATOR 279 3. The line selection circuit. 4. The trunk identifier and selection relay circuits. (For an understanding of how these frame circuits work through the line (link marker connector and markers in the No. 5 system, the reader should consult the references.) The common central office circuits will be described first. a. Scanner Pulse Generator The scanner pulse generator, shown in Fig. 24, produces continuously the combination of VG, VF and RS or reset pulses, described in connec- tion with Fig. 18(a), required to drive the scanners for a number of concentrators. The primary pulse source is a 1,000-cycle transistor oscillator. This oscillator drives a transistor bi-stable circuit arranged as a binary counter such that on each cycle of the oscillator output it alternately assumes one of its states. Pulses produced by one state drive a 5-stage counter. Pulses produced by the other state through gates drive a 12-stage counter. The pulses which drive the 5-stage counter are the same pulses which are used for the VF pulses to drive scanners. Each time the first stage of the 5-stage counter is on, a gate is opened to allow a pulse to drive the 12-stage counter. The pulses which drive the 12-stage counter are also the pulses used as the VG pulses for driving the scanners. They are out of phase with the VF pulses. When the last stage of the 12-stage counter is on, the gate which r VFC Fig. 24 — Scanner pulse generator. 280 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195(5 transmits pulses to the 12-stage counter is closed and another gate is opened which produces the reset pulse. The reset pulse is thereby trans- mitted to the scanners in place of the first vertical group pulse. At the same time the 5 and 12-stage counters in the scanner pulse generator are reset to enable the starting of a new cycle. In the central office control circuits, out of phase pulses on lead TP similar to those which drive the VG counters at the concentrator are used for various gating operations. b. The Originating Call Detection and Line Number Registration The originating call detector (Fig. 25) and the originating line num- ber register (Fig. 26) together receive the information from the line concentrator used to identify the number of the line making a service i request. The receipt of the service request pulse from a concentrator i in a particular time slot will set a transistor bi-stable circuit HGT of { Fig. 25 associated with that concentrator if no other originating call is being served by the frame circuits at* this time. The originating line number register consists of a 5 and 12-stage counter. These counters are normally driven through gates in syn- chronism with the scanning counters at concentrators with pulses sup- plied from the scanner pulse generator. When a service request pulse is received from any of the concentrators served by a line link frame, a pulse is sent to the originating line number register which operates a bi-stable circuit over a lead RH in Fig. 26. This bi-stable circuit then closes the gates through which the 5- and 12-stage counters are being driven, and also closes a gate which prevents them from being reset. TO TRAFFIC I RECORDER I TO ORIGINATING CALL REGISTER I TO CONCENTRATOR I CONTROL TRUNK Fig. 25 — Originating call detector. EEMOTE CONTROLLED LINE CONCENTRATOR 281 In this way, the number of the line which originated a service request is locked into these counters until the bi-stable circuit is restored to nor- mal. The HGT bi-stable circuit of Fig. 25 indicates which particular con- centrator has originated a service request. A relay in the collector cir- cuit has contacts which pass this information on to the other central office control circuits to indicate the number of the concentrator on the frame which is requesting service. This is the same as a horizontal group on a regular line link frame and hence the horizontal group designation is used to identify a concentrator. With the operation of this relay, relays associated with the counters of the originating line number register are operated. These relays indicate to the other central office circuits the vertical file and vertical group identification of the calling line. Contacts on the vertical group relays are used to set a bi-stable circuit associated with lead RL of Fig. 25 each time the scanner pulse generator generates a pulse corresponding to the vertical file of the calling line number registered. The operation of the HGT bi-stable circuit inhibits in the concentra- tor control trunk circuit (Fig. 27) the transmission of further VG and SRS FROM CONCENTRATOR CONTROL TRUNK CIRCUIT RB RH RH FROM SCANNER PULSE GENERATOR VF VFO-4 VG RS 1 *" 5-STAGE COUNTER ^ 12-STAGE COUNTER ^ ^ Fig. 26 — Originating line number register. 282 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 reset pulses to the concentrator so that, as described in Section 7a, only the VF counter continues to step once each 0.010 sec. So long as the line continues to request service this service request pulse is gated to reset the RL bi-stable circuit within the same time slot that it was set. If, however, a request for service is abandoned the RL bi-stable cir- cuit of Fig. 26 will remain on and permit a TP pulse from the scanner pulse generator to reset the HGT bi-stable circuit which initiated the service request action. Whenever the RH bi-stable circuit of Fig. 26 is energized it closes a gate over lead SRS for each concentrator to prevent any further service request pulses from being recognized until the originating call which has been registered is served. The resetting of the RH bi-stable circuit occurs once the call has been served. When more than one line concen- trator is being served it is possible that the HGT bi-stable circuit of more than one concentrator will be set simultaneously as a result of coincidence in service requests from correspondingly numbered lines in these concentrators. The decision as to which concentrator is to be served is left to the marker, as it would normally decide which horizontal group to serve. c. Line Selection On all calls, originating and terminating, the marker transmits to the frame circuits the complete identity of the line which it will serve. In the case of originating calls it has received this information in the manner described in Section 8b. In either case, it operates wire spring relays VGO-U and VFO-4, which enable gates so that the information may be stored in the 5- and 12-stage counters of the line selection circuit shown " in Fig. 28. The process of reading into the line selection counters starts when selection information has been received by the actuation of the HGS bi-stable circuit in the concentrator control trunk circuit of Fig. 27. This action stops the regular transmission of scanner pulses if they have not been stopped as a result of a call origination. At the same time it enables gates for transmission of information from the line selection circuit. Fig. 28. The ST bi-stable circuit of the line selection circuit is also enabled to start the process of setting the line selection counters. The next TP pulse sets the Rl bi-stable circuit. This bi-stable circuit enables a gate which permits the next TP pulse to set the counters and transmit a re- set pulse to the concentrator through pulse amplifier RIA. At the same time bi-stable circuit ST is reset to prevent the further read-in cr reset \ REMOTE CONTROLLED LINE CONCENTRATOR 283 pulses and to permit pulses through amplifier OPA to start the out- pulsing of line selections. These pulses pass to the VGP and VFP leads as long as the VG and VF line selection counters have not reached their first and last stages respectively. The output pulses to the con- centrator are also fed into the drive leads of these counters so that, as the counters in the concentrator are stepped up, the counters in the central office line selection circuit are stepped down. When the first stage of the VF counter goes on, the VF pulses are no longer transmitted until the first stage of the VG counter goes on. This insures that a VF pulse is the last to be transmitted. Also this pulse is not transmitted until the other frame circuits have successfully completed selections of an idle concentrator trunk. Then bi-stable circuit VFLD is energized, TO ORIGINATING CALL DETECTOR I VF- FROM VG U SCANNER PULSE GENERATOR FROM LINE [-SELECTION CIRCUIT Fig. 27 — Concentrator control trunk circuit. 284 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 producing, during its transition, the last VF pulse for transmission to the concentrator. d. Trunk Selection and. Identification The process of selecting an idle concentrator trunk to which the line has access utilizes familar relay circuit techniques.^^ This circuit, in Fig. 29, will not be described in detail. One trunk selection relay, TS, is operated indicating the preferred idle trunk serving a line in the particu- lar vertical group being selected as indicated by the VG relay which has been operated by the marker. The TS4 and TS5 relays select trunks 8 and 9 which are available to each line while the 4 trunks available to only half of the lines are selected by relays TS0-TS3. The busy or idle condition of each trunk is indicated by a contact on the hold magnet associated with each trunk through TRUNK SELECTION COMPLETE T VFLD _l O cr I- z Oh ^5 tr u O cr Z : LU : ^! o o o t- VFP VGP_ RS VFLI VFL 2 0-1 5-STAGE COUNTER VF4X -2V VGO y^ 12-STAGE COUNTER 0-0 ST OPA R1A ST FROM SCANNER PULSE GENERATOR I TP Fig. 28 — Line selection circuit. REMOTE CONTROLLED LINE CONCENTRATOR 285 relay HG which operates on all originating and terminating calls to the particular concentrator served by these trunks. The end chain relay TC of the lockout trunk selection circuit^^ connects battery from the SR relay windings of idle trunks to the windings of the TS relays to permit one of the latter relays to operate and to steer circuits, not shown on Fig. 29, to the hold magnet of the trunk and to the tip-and-ring con- ductors of the trunk to apply the selection voltages shown on Figs. 13 and 14. The path for operating the hold magnet originates in the marker. The path looks like that which the marker uses on the line hold mag- net when setting up a call on a regular line link frame. For this reason and other similar reasons this concentrator line link frame concept has been nicknamed the "fool-the-marker" scheme. Should a hold magnet release while a new call is being served the ground from the TC relaj^ normal or the TS relay winding holds relay CONCENTRATOR TRUNK SWITCH CROSSPOINTS SR [ LINE LINK NUMBER FROM MARKER I -48 V Fig. 29 — Trunk selection and identification. 286 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 SR operated through its own contact until the new call has been set up. This prevents interference of disconnect pulses applied to the trunk when a selection is being made and insures that a disconnect pulse is transmitted before the trunk is reused. A characteristic of the No. 5 crossbar system is that the originating connection to a call register including the line hold magnet is released and a new connection, known as the "call back connection", is estab- lished to connect the line to a trunk circuit after dialing is completed. With concentrator operation the concentrator trunk switch connection is released but the disconnect signal is not sent to the concentrator as a result of holding the SR relay as described above. However, the marker does not know to which trunk the call back connection is to be estab- lished. For this reason the frame circuits include an identification proc- ess for determining the number of the concentrator trunk to be used on call back prior to the release of the originating register connection, i Identification is accomplished by the marker transmitting to the frame circuits the number of the link being used on the call. This in- formation is already available in the No. 5 system. The link being used is marked with —48 volts by a relay selecting tree^" to operate the TS relay associated with the trunk to which the call back connection is to be established. Relay CB (Fig. 29) is operated on this type of call in- stead of relay HG. The circuits for reoperating the proper hold magnet are already available on the TS relay which was operated, thereby rc- selecting the trunk to which the customer is connected. The concen- trator connection is not released when the hold magnet releases and again the marker operates as it would on a regular line link frame call. 9. FIELD TRIALS Three sets of the experimental equipment described here have been constructed and placed in service in various locations. The equipment for these trials is the forerunner of a design for production which will incorporate device, circuit and equipment design changes based on the trial experiences. Fig. 30 shows the cabinet mounted central office trial equipment with the designation of appropriate parts. For the field trials described, the line links on a particular horizontal level of existing line link frames were extended to a separate cross-bar switch provided for this purpose in the trial equipment. The regular line link connector circuits were modified to work with the trial control circuits whenever a call was originated or terminated on this level. N(i lines were terminated in the regular primary line switches for this level. REMOTE CONTROLLED LINE CONCENTRATOR 287 10. MISCELLANEOUS FEATURES OF TRIAL EQUIPMENT There are a number of auxiliary circuits provided with the trial equip- ment to aid in the solutions of problems brought about by the concepts of concentrator service. One of the purposes of the trials was to deter- mine the way in which the various traffic, plant and commercial ad- CONCENTRATOR TRUNK SWITCH SERVICE OBSERVING TEST CONTROL-] SIMULATOR TRUNK DISCONNECT RELAYS CONCENTRATOR CONNECTOR RELAYS FRAME RELAY CIRCUITS SERVICE DENIAL FRAME ELECTRONIC CIRCUITS POWER SUPPLY LINE CONDITION TESTER Fig. 30 — Trial central office equipment. 288 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ministrative functions could be economically performed when concen- trators become common telephone plant facilities. The more important of these miscellaneous features are discussed under the following head- ings : a. Traffic Recording J To measure the amount and characteristics of the traffic handled by the concentrator a magnetic tape recorder, Fig. 31, was provided for each trial. The number of the lines and trunks in use each 15 seconds during programmed periods of each day were recorded in coded form with polarized pulses on the 3-track magnetic tape moving at a speed of 1}/2" per second. Combinations of these pulses designate trunks busy on intra-concentrator connections and reverting calls. The line busy indications were derived directly from the line busy information received during regular scanning at the concentrator. Dur- ing one cycle in each 15 seconds new service requests were delayed to insure that a complete scan cycle would be recorded. Terminating calls were not delayed since marker holding time is involved. Trunk condi- tions are derived for a trunk scanner provided in the recorder. In addition to recording the line and trunk usage, recordings were made on the tape for each service request detected during a programmed period to measure the speed with which each call received dial tone and the manner in which the call was served. In this type of operation the length of the recording for each request made at a tape speed of only \i!' per second is a measure of service delay time. As may be observed from Fig. 31 the traffic recorder equipment was built with vacuum tubes and hence required a rather large power supply. It is expected that a transistorized version of this traffic recorder serv- ing all concentrators in a central office will be included in the standard model of the line concentrator equipment. With this equipment, traffic engineers will know more precisely the degree to which each concentra-- tor may be loaded and hence insure maximum utilization of the concen- trator equipment. b. Line Condition Tester It has been a practice in more modern central office equipment to include automatic line testing equipment.^^ An attempt has been made to include similar features with the concentrator trial equipment. The line condition tester (see Fig. 30) provides a means for automatically connecting a test circuit to each line in turn once a test cycle has been I REMOTE CONTROLLED LINE CONCENTRATOR 289 ! , P ] 'f ^ u POWER SUPPLIES AND PROGRAMMER I Fig. 31 — Traffic recorder. 290 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 manually initiated. This test is set up on the basis of the known concen- trator passive line circuit capabilities. Should a line fail to pass this test, the test circuit stops its progress and brings in an alarm to summon central office maintenance personnel. The facilities of the line tester are also used to establish, under manual control, calls to individual lines as required to carry out routine tests. c. Simulator As the central office sends out scanner control pulses either no signal, a line busy or service request pulse is returned to the central office in each time slot. The simulator test equipment, shown in Fig. 30, was designed to place pulses in a specific time slot to simulate a line under test at the concentrator. In addition to transmitting the equivalent of concentrator output pulses the simulator can receive the regular line selection pulses trans- mitted to the concentrator for purposes of checking central office opera- tions. It is possible by combined use of the line tester and simulator to observe the operation of the concentrator and to determine the probable cause when a fault occurs. d. Service Observing The removal of the line terminals from the central office poses a num- ber of problems in conjunction with the administration of central office equipment. One of these is service observing. To maintain a check on the quality of service being rendered by the telephone system, service observing taps are made periodically on tele- phone lines. This is normally done by placing special connector shoes on line terminations in the central office. To place such shoes at the remote concentrator point would lead to administrative difficulties and added expense. Therefore, a method was devised to permit service observing equipment to be connected to con- centrator trunks on calls from specific lines which were to be observed. This mcithod consisted of manual switches on which were set the number of the line to be observed in terms of vertical group and vertical file. Whenever this line originated a call and the call could be placed over the first preferred trunk, automatic connection was made to the service ob- serving desk in the same manner as would occur for a line terminated directly in the central office. In addition, facilities were provided for trying a new service observ- ing technique where calls originating over a particular concentrator REMOTE CONTROLLED LINE CONCENTRATOR 291 trunk would be observed without knowledge of the originating line num- ber. For this purpose a regular line observing shoe was connected to one of the ten concentrator trunk switch verticals in the trial equipment and from here connected to the service observing desk in the usual manner. The basic service observing requirements in connection with line concentrator operation have not as yet been fully determined. How- ever, it appears at this time that the trunk observing arrangement may be preferable. e. Service Denial In most systems denial of originating service for non-payment of telephone service charges, for trouble interception and for permanent signals caused by cable failures or prolonged receiver-off-hook conditions may be treated by the plant forces at the line terminals or by blocking the line relay. To avoid concentrator visits and to enable the prompt clearing of trouble conditions which tie up concentrator trunks, a ser- vice denial feature has been included in the design of the central office circuits. This feature consists of a patch-panel with special gate cords which respond to particular time slots and inhibit service request signals pro- duced by a concentrator during this period. In this way service requests can be ignored and prevent originating call service on particular lines until a trouble locating or other administrative procedure has been invoked. f. Display Circuit A special electronic switch was developed for an oscilloscope. This arrangement permited the positioning of line busy and service request pulses in fixed positions representing each of the 60 lines served. Line busy pulses were shown as positive and service request pulses as negative. This plug connected portable aid, see Fig. 32, was useful in tracing calls and identifying lines to which service may be denied, due to the existence of permanent signals. Other circuits and features, too detailed to be covered in this paper, have been designed and used in the field trials of remote line concen- trators. Much has been learned from the construction and use of this equipment which will aid in making the production design smaller, lighter, economical, serviceable and reliable. Results from the field trials have encouraged the prompt undertaking 292 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Fig. 32 — Pulse display oscilloscope. REMOTE CONTROLLED LINE CONCENTRATOR 293 of development of a remote line concentrator for quantity production. The cost of remote line concentrator equipment will determine the ul- timate demand. In the meantime, an effort is being made to take advan- tage of the field trial experiences to reduce costs commensurate with insuring reliable service. The author wishes to express his appreciation to his many colleagues at Bell Telephone Laboratories whose patience and hard work have been responsible for this new adventure in exploratory switching de- velopment. An article on line concentrators would not be complete without mention of C. E. Brooks who has encouraged this development and under whose direction the engineering studies were made. BIBLIOGRAPHY 1. E. C. Molina, The Theory of Probabilities Applied to Telephone Trunking Problems, B.S.T.J., 1, pp. 69-81, Nov., 1922. 2. Strowger Step-bv-Step System, Chapter 3, Vol. 3, Telephone Theory and Practice by K.B. Miller. McGraw-Hill 1933. 3. F. A. Korn and J. G. Ferguson, Number 5 Crossbar Dial Telephone Switching System, Elec. Engg., 69, pp. 679-684, Aug., 1950. 4. U.S. Patent 1,125,965. 5. O. Myers, Common Control Telephone Switching Systems, B.S.T.J., 31, pp. 1086-1120, Nov., 1952. 6. L. J. Stacy, Calling Subscribers to the Telephone, Bell Labs. Record, 8, pp. 113-119, Nov., 1929. 7. J. Meszar, Fundamentals of the Automatic Telephone Message Accounting System, A. I. E. E. Trans., 69, pp. 255-268, (Part 1), 1950. 8. O. M. Hovgaard and G. E. Perreault, Development of Reed Switches and Relays, B.S.T.J., 34, pp. 309-332, Mar., 1955. 9. W. A. Malthaner and H. E. Vaughan, Experimental Electronically Controlled Automatic Switching System, B. S.T.J., 31, pp. 443-468, May, 1952. 10. S. T. Brewer and G. Hecht, A Telephone Switching Network and its Electronic Controls, B.S.T.J., 34, pp. 361-402, Mar., 1955. 11. L. W. Hussey, Semiconductor Diode Gates, B.S.T.J., 32, pp. 1137-54, Sept., 1953. 12. U. S. Patent 1,528,982. 13. J. J. EbersandS. L. Miller, Design of Alloyed Junction Germanium Transis- tor for High-Speed Switching, B. S.T.J. , 34, pp. 761-781, July, 1955. 14. W. B. Graupner, Trunking Plan for No. 5 Crossbar System, Bell Labs. Record, 27, pp. 360 365, Oct., 1949. 15. G. L. Pearson and B. Sawyer, Silicon p-n Junction Alloy Diodes, I.R.E. Proc, 42, pp. 1348-1351, Nov." 1952. 16. A. E. Anderson, Transistors in Switching Circuits, B.S.T.J., 31, pp. 1207- 1249, Nov., 1952. 17. J. J. Ebers and J. L. Moll, Large-Signal Behavior of Junction Transistors, I. R. E. Proc, 42, pp. 1761-1784, Dec, 1954. 18. J. J. Ebers, Four-Terminal p-n-p-n Transistors, I. R. E. Proc, 42, pp. 1361- 1364, Nov., 1952. 19. A. E. Joel, Relay Preference Lockout Circuits in Telephone Switching, Trans. A. L E. E., 67, pp. 720-725, 1948. 20. S. H. Washburn, Relay "Trees" and Symmetric Circuits, Trans. A. I. E. E., 68, pp. 571-597, 1949. 21. J. W. Dehn and R. W. Burns, Automatic Line Insulation Testing Equipment for Local Crossbar Systems, B.S.T.J., 32, pp. 627-646, 1953. Transistor Circuits for Analog and Digital Systems* By FRANKLIN H. BLECHER (Manuscript received November 17, 1955) This paper describes the application of junction transistors to precision circuits for use in analog computers and the input and output circuits of digital systems. The three basic circuits are a summing amplifier, an inte- grator, and a voltage comparator. The transistor circuits are combined into a voltage encoder for translating analog voltages into equivalent time inter- vals. 1.0. INTRODUCTION Transistors, because of their reliability, small power consumption, and small size find a natural field of application in electronic computers and data transmission systems. These advantages have already been realized by using point contact transistors in high speed digital com- puters. This paper describes the application of junction transistors to precision circuits which are used in dc analog computers and in the input and output circuits of digital systems. The three basic circuits which are used in these applications are a summing amplifier, an inte- grator, and a voltage comparator. A general procedure for designing these transistor circuits is given with particular emphasis placed on new design methods that are necessitated by the properties of junction transistors. The design principles are illustrated by specific circuits. The fundamental considerations in the design of transistor operational amplifiers are discussed in Section 2.0. In Section 3.0 an illustrative summing amplifier is described, which has a dc accuracy of better than one part in 5,000 throughout an operating temperature range of to 50°C. The feedback in this amplifier is maintained over a broad enough frequency band so that full accuracy is attained in about 100 micro- seconds. The design of a specific transistor integrator is presented in Section * Submitted in partial fulfillment of the requirements for the degree of Doctor of Electrical Engineering at the Polytechnic Institute of Brooklyn. 295 296 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 I 4.0. The integrator can be used to generate a voltage ramp which is linear to within one part in 8,000. By means of an automatic zero set (AZS) circuit which uses a magnetic detector, the slope of the voltage ramp is maintained constant to within one part in 8,000 throughout a temperature range of 20°C to 40°C. The voltage comparator, described in Section 5.0, is an electrical de- vice which indicates the instant of time an input voltage waveform passes through a predetermined reference level. By taking advantage of the properties of semiconductor devices, the comparator can be de- signed to have an accuracy of ±5 millivolts throughout a temperature range of 20°C to 40°C. In Section 6.0, the system application of the transistor circuits is demonstrated by assembling the summing amplifier; the integrator, and the voltage comparator into a voltage encoder. The encoder can be used J to translate an analog input voltage into an equivalent time interval with an accuracy of one part in 4,000. This accuracy is realized through- out a temperature range of 20°C to 40°C for the particular circuits described. 2.0. FUNDAMENTAL CONSIDERATIONS IN THE DESIGN OF OPERATIONAL AMPLIFIERS The basic active circuit used in dc analog computers is a direct coupled negative feedback amplifier. With appropriate input and feedback net- works, the amplifier can be used for multiplication by a constant coef- ficient, addition, integration, or differentiation as shown in Figure 1 The accuracy of an operational amplifier depends only on the passive components used in the input and feedback circuits provided that there is sufficient negative feedback (usually greater than 60 db). The time that is required for the amplifier to perform a calculation is an inverse f miction of the bandwidth over which the feedback is maintained. Thus a fundamental problem in the design of an operational amplifier is the development of sufficient negative feedback over a reasonably broad frequency range. The associated problem is the realization of satisfactory stability margins. Finally there is the problem of reducing the drift which is inherent in direct coupled amplifiers and particularly troublesome for transistors because of the variation in their character- istics with temperature. The first step in the design is the blocking out of the configuration for the forward gain circuit (designated A in Fig. 1). Three primary re- quirements must be satisfied: (1) Stages must be direct coupled. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 297 (2) Amplifier must provide one net phase reversal. (3) Amplifier must have enough current gain to meet accuracy re- quirements. Three possible transistor connections are available: (a) the common base connection which may be considered analogous to the common grid vacuum tube connection; (b) the common emitter connection which is analogous to the common cathode connection; and (c) the common collector connection which is analogous to the cathode follower connection. These three configurations together with their approximate equivalent circuits are shown in Fig. 2. It has been shown^ that for most junction transistors the circuit element a is given by the expression a = sech W (1 + PTrn) 1/2 (1)^ where W is the thickness of the transistor base region, Lm is the diffusion length and t„, the lifetime of minority charge carriers in the base region, Rk I — ^AV E- "J -^ Wvr Eo Rk A/3EL Rk ^ •=0" Rj (i-A/i)"^ Rj ^l (a) MULTIPLICATION BY A CONSTANT COEFFICIENT E, R. E ^2 E3 ^^ Rk I — vv\- Eo = E Rk A/bEj p, Rj (i-A/3) (b) ADDITION N r- . •RKEf: c §i — vw- £[Eo] A/3 £[el] sl[eQ Eo ^N^^^^?|^-PH«[EJ (d) DIFFERENTIATION (l-A/3) pRC ~ pRC (C) INTEGRATION note; £[Eo] = LAPLACE TRANSFORM OF OUTPUT VOLTAGE £[Ei1 = LAPLACE TRANSFORM OF INPUT VOLTAGE p = jco Fig. 1 — Summary of operational amplifiers. * This expression assumes that the injection factor y and the collector efficiency at are both unity. This is a good approximation for all alloy junction transistors and most grown junction transistors. 298 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 and 7? — ju. At frequencies less than Ua/^ir, (1) can be approximated by a = 1 + ^ (2)' COa where ao is the low frequency value of a ^ 1 l/TT 2U. and 2.4Z). CCa = w (Dm is the diffusion constant for the minority charge carriers in the base region). A readily measured parameter called alpha (a), the short circuit current gain of a junction transistor in the common base connec- SCHEMATIC Zc = EQUIVALENT CIRCUIT s ^ — *■ e/ (V b =^ V\V aZcLe Tb (a) COMMON BASE Lb : rb Zed -a) aZc'Lb 'X, ■re (b) COMMON EMITTER i^b rb aZc re aZcLb Zed -a) (C) COMMON COLLECTOR re 1 + prcCc a P — ao i-hP re = COLLECTOR RESISTANCE Cc = COLLECTOR CAPACITANCE ZTT ALPHA-CUTOFF FREQUENCY Fig. 2 — Basic transistor connections. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 299 tion, is related to a by the equation aZe -{■ n /^x Ze + n For most junction transistors the base resistance, n , is much smaller than the collector impedance | Zc |, at frequencies less than Wa/27r. There- fore, a ^ a and Ua/^ir is very nearly equal to the alpha-cutoff frequency, the frequency at which | a | is down by 3 db. The transistor parameters r^ and n are actually frequency sensitive and should be represented as impedances. However, good agreement between theory and experiment is obtained at frequencies less than Wa/27r with re and n assumed constant. The choice of an appropriate transistor connection for a direct coupled, negative feedback amplifier, is based on the following reasoning. The common base connection may be ruled out immediately because this connection does not provide current gain unless a transformer interstage is used. The common emitter connection provides short circuit current gain and a phase reversal for each stage. Thus if the amplifier is com- posed of an odd number of common emitter stages, all three requirements previously listed, are satisfied. A common emitter cascade has the addi- tional practical advantage, that by alternating n-p-n and p-n-p types of transistors, the stages can be direct coupled with practically zero inter- stage loss. The common collector connection provides short circuit current gain but no phase reversal. Consequently, the dc amplifier cannot consist entirely of common collector stages and operate as a negative feedback amplifier. This paper will consider only the common emitter connection since, in general, for the same number of transistor stages, the common emitter cascade provides more current gain than a cascade composed of both common collector and common emitter stages. 2.1 Evaluation of External Voltage Gain Since the equivalent circuit of the junction transistor is current acti- vated, it is convenient to treat feedback in a single loop transistor ampli- fier as a loop current transmission (refer to Appendix I) instead of as a loop voltage transmission which is commonly used for single loop vacuum tube amplifiers.^ Fig. 3 shows a single loop feedback amplifier in which a fraction of the output current is fed back to the input. A is defined as the short circuit current gain of the amplifier without feedback, and jS is defined as the fraction of the short circuit output current (or Norton 300 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 equivalent circuit current) fed back to the input summing node. With these definitions, he = A/in' (4) la - 131. sc where /sc is the Norton equivalent short circuit current. From Kirchhoff's first law /in = /in + Iff Combining relations (4) to (6) yields 'sc A (5) (6) (7) /in 1 - A^ Expression (7) provides a convenient method for evaluating the external ^ [|N I IN > ^ Fig. 3 — Single loop feedback amplifier. voltage gain of an operational amplifier. Fig. 4 shows a generalized op- erational amplifier with N inputs. With this configuration, IN j=i L TTT he r, I Zj (S) where Ej , j = 1,2, • ■ • , N, are the N input voltages referred to the ground node. Zj,j— 1,2, • ■ ■ , N, are the A^ input impedances ZiN is the input impedance of the amplifier measured at the summing node with the feedback loop opened. Eo //i sc UT 'IN la = A Eovr = Zk /sc ~ / (3 Rl Zovt (5>) (10) TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 301 where Zovr is the output impedance of the amplifier measured with the feedback loop opened. The expression for the output voltage is obtained by combining (7), (8), (9), and (10). E, OUT N y = zL ^i 7" ;=1 ^i A^ + 3 = 1 ^1 _ (iir where A^ = A 1 - 'IN \Ri + /OUT 1 _^ ^ + ^^^ Rl Zovr IA/3 is equal to the current returned to the summing node when a unit Ei Z, MN 1/3 Zk I IN Zn Zls J7 1/5 Zk Equt NORTON EQUIVALENT CIRCUIT Fig. 4 — Generalized operational amplifier. icurrent is placed into the base of the first transistor stage (/in = 1). If I A^ 1 is much greater than ] Zj^'/Zr \ and 1 + L 'IN then N Eqvt — ~ 2^ J^j nT (12) y=i The accuracy of the operational amplifier depends on the magnitude of AjS and the precision of the components used in the input and feedback networks as can be seen from (11). There is negligible interaction between the input voltages because the input impedance at the summing node is equal to Zin' divided by (1 — A^)? This impedance is usually negligibly tsmall compared to the impedances used in the input circuit. * In general, E,- and Eout are the Laplace transforms of the input and output fvoltages, respectively. 302 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 2.2. Methods Used to Shape the Loop Current Transmission An essential consideration in the design of a feedback amplifier is the provision of adequate margins against instability. In order to accomplish this objective, it is necessary to choose a criterion of stability. In Ap- pendix I it is shown that it is convenient and valid to base the stability of single loop transistor feedback amplifiers on the loop current trans- mission. In order to calculate the loop current transmission of the dc amplifier, the feedback loop is opened at a convenient point in the cir- cuit, usually at the base of one of the transistors, and a unit current is injected into the base (refer to Fig. 24). The other side of the opened loop is connected to ground through a resistance (r^ -j- r^) and voltage Veli • In many instances, the voltage re/4 can be neglected. If | Zj? | and 3=1 Zj I are much greater than | Z IN then A/3 is very nearly equal to the loop ciu'rent transmission. For absolute stability^ the amplitude of the loop current transmission must be less than unity before the phase shift (from the low frequency value) exceeds 180°. Consequently, this charac- teristic must be controlled or properly shaped over a wide frequency 10 _J LU O LU Q < o \- z LU a. o 40 U), a;,' Wa ^{\-\-S)u)^ \ ao ^" ■\ \, \J i "^ \ 30 1- ao+cT t- — . ao \ 7~' \ \ ^ AM PL ITUC E \ \ 20 10 i-ao + 7_ AMPLITUDE' (WITH LOCAL FEEDBACK) \ \ \ \ S < PHASE (WITH LOCAL FEEDBACK) phase\ \ y \ > \ \ ao ^'^ f' \+S 1 -270° X — . s \ \ N d «/c -10 20 30 40 "^ N \ ^ ^ \ ^ \ \ •180 -200 -220 •240 •260 uj _l 2 < -280 •300 LU < I I Q- -320 ■340 ,02 2 5 ,q3 2 ^ ,o4 •=; S ,q5 t S ,q6 5 ■/^4 2 5 ,„s 2 5 ,„6 2 5 ^q7 2 5 jq8 FREQUENCY IN CYCLES PER SECOND Fig. 5 — Current transmission of a common emitter stage. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 303 band. In addition, it is desirable that the feedback fall off at a rate equal to or less than 9 db per octave in order to insure that the dc aniplifier has a satisfactory transient response. Three methods of shaping are described in this paper; local feedback shaping, interstage network shaping, and (3 circuit shaping. Local feed- back shaping will be described first. The analysis starts by considering the current transmission of a common emitter stage, ecjuivalent circuit shown in Fig. 2(b). If the stage operates into a load resistance Rl , then to a good approximation the current transmission is given by where Gr = r" = ^ ~ ^° +/ (13)^ ^^ 1 + ^ + '^ wi a)aCOc(l — ao -\- 8) RL+Te 8 = COl = (1 - ao + 8) 1 + 5,1 -^ ^ alpha-cutoff frequency Ztt 1 Uc (7?x, + re)Cc It is apparent from expression (13) that if (1 — Oo + 8) is less than 0.1, then the current gain of the common emitter stage falls off at a rate of 6 db per octave with a corner frequency at wi .f A second 6 db per octave cutoff with a corner frequency at [co^ + (1 + 5)aJc] is introduced by the p" term in the denominator of (13). A typical transmission characteristic is shown in Fig. .5. The current gain of the common emitter stage is unity at a frequency equal to ao 1 +5 I 1 * Expressions (13) and (14) are poor approximations at frequencies above ' coo/27r. ' t Strictly speaking the corner frequency is equal to 01/2 tt. However, for sim- plicity, corner frequencies will be expressed as radian frequencies. 304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Since the phase crossover of A|S* is usually placed below this frequency, the principal effect of the second cutoff is to introduce excess phase. This excess phase can be minimized by operating the stage into the smallest load resistance possible, thus maximizing Wc . j An undesirable property of the common emitter transmission charac- teristic is that the corner frequency coi occurs at a relatively low fre- quency. However, the corner frequency can be increased by using local feedback as shown in Fig. 6(a). Shunt feedback is used in order to pro- vide a low input impedance for the preceding stage to operate into. The amplitude and phase of the current transmission is controlled prin- cipally by the impedances Z\ and Z2 . If | A& \ is much greater than one, and if /3 ;^ ^1/^2 , then from (7) the current transmission of the stage is approximately equal to — Z2/Z1 . Because of the relatively small size of A^ for a single stage, this approximation is only valid for a very limited range of values of Zi and Z2 . If Zi and Zi are represented as resistances R\ and Ri , then the current transmission of the circuit is given to a good approximation by tto h. _ R2 1 — gp + 7 ^^ = /i= ~{R2 + n)r_^p_^ v' where 7 = coi = COc = Co/ COaCOcCl — Oo + 7), R\ + Te _, Rl + Te R2 + ^6 I (14) (/?2 + rb)rc i22 + n (1 +ao + ro + 7) 1 + 7 1 {R, -f re)Cc i By comparing (14) with (13), it is evident that the negative feedback has reduced the low-frequency current gain from ao/(l — ao) (5 may usually be neglected) to ( R2 \ I «0 \ _ , ^2 R2 + rj \1 - ao + 7/ ^1 + re (if 7 > 1 - ao) .-•! * The phase crossover of A/3 is equal to the frequency at which the phase shift of A/3 from its low-frequency value is 180°. I TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 305 The half power frequency, however, has been increased from 1— Oo , 1— Oo + 7 t:^ 1 + 7 , 1 as shown by the dashed curves in Fig. 5.* The bandwidth of the common emitter stage can be increased without reducing the current gain at dc and low-frequencies by representing Zi by a resistance Ri , and Z2 by a resistance R2 in series with a condenser C2 . If I/R2C2 is much smaller than co/, then the current transmission of the stage is given by (14) multiplied by the factor P 1 + C04 P (15) where 602 Wi H^^i 1 - cro + Ri + re C2(/?2 + r6)(l - ao + 7) The current transmission for this case is plotted in Fig. 6(b). The con- denser d introduces a rising 6 db per octave asymptote with a corner frequency at wi . At dc the current gain is equal to ao 1 — ao + 5 A second method of shaping the loop current transmission char- acteristic of a feedback amplifier is by means of interstage networks. These networks are usually used for reducing the loop current gain at relatively low frequencies while introducing negligible phase lag near the gainf and phase crossover frequencies. Interstage networks should be designed to take advantage of the variable transistor input impedance. The input impedance of a transistor in the common emitter connection * In Figs. 5 and 6(b), the factor R^/iRi + n) is assumed equal to unity. This is ' a good approximation since in practice R2 is equal to several thousand ohms while rt is equal to about 100 ohms. t The gain crossover frequency is equal to the frequency at which the magni- tude of Al3 is unity. 306 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 is given by the expression 'INP UT = ?'6 + ^e(l — Gi) (16)1 where Gj is the current transmission given by (13). If Gi at dc is much' greater than 1, then the input impedance and the current transmission of the common emitter stage fall off at about the same rate and with approximately the same corner frequency (wi). The input impedance finally reaches a limiting value equal to r^ + Vb . A particularly useful interstage network is shown in Fig. 7(a). This network is analyzed in Appendix II and Fig. 7(b) shoAvs a plot of the 60 50 40 30 20 Z < z UJ 10 tr cr D U -10 -20 -30 (a: EQUIVALENT CIRCUIT \ \ (b) \ AMPLITUDE an \ (WITHOUT LOCAL \ FEEDBACK) 1-ao+d" ^ - ^" \ ■*•> 1 ^ s .AMPLITUDE ^4 ^>CiL ^'' ,^_ •— ^ ■ ■~^ r"**^ cvz r^ i / "^v ^ ' 1 >^ / V \ \ \ X V ^ ao i-ao+ 7 - / A / ^ \ s. \ k \ \ \ / / PH/ >s> / \ \ \ \ V \ \ PHASE N \ (WITHOUT LOCAL \ s FEEDBACK) s w k. - ^-. ■"••^^, 120 140 -160 10 UJ m cr -180 liJ Q Z -200 ^ z < -220 , 10 2 5p2 5,2 5.2 5,2 5 -!•= in^ m^ in5 10'= lO-^* 10^ 10= FREQUENCY IN CYCLES PER SECOND lO'' -240 ' 260 - -280 10' Fig. 6 — Negative feedback applied to a common emitter stage. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 307 resulting current transmission. The amplitude of the transmission falls off at a rate of 6 db per octave with the corner frequency C05 determined by C'3 and the low frequency value of the transistor input impedance. The inductance L3 introduces a 12 db per octave rising asymptote with a corner frequency at C03 = WLsCs . The corner frequencies C03 and C05 are selected in order to obtain a desirable loop current transmission characteristic (specific transmission characteristics are presented in Sec- tions 3.0 and 4.0). The half power frequency of the current transmission of the transistor, wi , does not. appear directly in the transmission char- acteristic of the circuit because of the variation in the transistor input impedance with frequency. The overall (3 circuit of the feedback amplifier can also be used for i-ao+(J I ^ s LU Q z I - <l< z < 15 Z UI cc D u Q UJ y < 2 a o z 40 20 -20 -40 -60 -80 (b) / / CU5 u ■^3/ y' * ^^^ A^ ^ ^s^ 1 \. ^N, / ^s^ S^ / \ \ AMPLITUDE / / \ \^ / \ \ \ \ \ > X X V 1 1 / / / 1 1 1 1 1 _ \ \ \ \ \ s,. / cu,(rb+ ' Te \ \-do+l W \ .PHASE ^ Tb+le-l-Ra-K^iLa ^^ / \. *^.., — ^** X — - N - -135 10 LU UJ isog z < -225 1}^ < I a. -270 102 " "^ \0^ " = 10^ -^ = 105 FREQUENCY IN CYCLES PER SECOND Fig. 7 — Interstage shaping network. lO'' 308 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 shaping the loop current transmission. If the feedback impedance Zk (Fig. 4) consists of a resistance Rk and condenser Ck in parallel, then the loop current transmission is modified by the factor 1 + CO; 1 + ^ COS (17) where C07 = C08 = RkCk (Rl_±_Rk) RlRkC K Since Zk affects the external voltage gain of the operational amplifier, (11), the corner frequency C07 must be located outside of the useful fre- quency band. Usually it is placed near the gain crossover frequency in order to improve the phase margin and the transient response of the amplifier. In Sections 3.0 and 4.0, the above shaping techniques are used in the design of specific operational amplifiers. 3.0. THE SUMMING AMPLIFIER 3.1. Circuit Arrangement The schematic diagram of a dc summing amplifier is shown in Fig. 8. From the discussion in Section 2.0 it is apparent that each common emitter stage will contribute more than 90 degrees of high-frequency phase lag. Consequently, while the magnitude of the low-frequency : feedback increases with the number of stages, this is at the expense of , the bandwidth over which the negative feedback can be maintained. It is possible to develop 80 db of negative feedback at dc with three common emitter stages. This corresponds to a dc accuracy of one part in 10,000. In addition, the feedback can be maintained over a broad enough band in order to permit full accuracy to be attained in about 100 microseconds. Thus it is evident that the choice of three stages repre- sents a satisfactory compromise between accuracy and bandwidth ob- jectives. The output stage of the amplifier is designed for a maximum power dissipation of 75 milliwats and maximum voltage swing of ±25 volts I TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 309 when operating into an external load resistance equal to or greater than 50,000 ohms. A p-n-p transistor is used in the second stage and n-p-n transistors are used in the first and third stages. This circuit arrangement makes it possible to connect the collector of one transistor directly to the base of the following transistor without introducing appreciable interstage loss. ''Shot" noise" and dc drift are minimized by operating the first stage at the relatively low collector current of 0.25 milliamperes. The 110,000-ohm resistor provides the collector current for the first stage, and the 4,700-ohm resistor provides 3.8 milliamperes of collector current for the second stage. The series 6,800-ohm resistor between the xcond and third stages, reduces the collector to emitter potential of the second stage to about 4.5 volts. The loop current transmission is shaped by use of local feedback ap- plied to the second stage, by an interstage network connected between the second and third stages, and by the overall (3 circuit. The 200-ohm resistor in the collector circuit of the second stage is, with reference to Fig. 6(a), Zi . The impedance of the interstage network can be neglected since it is small compared to 200 ohms at all frequencies for which the local feedback is effective. The interstage network is connected between the second and third stages in order to minimize the output noise voltage. ^^'ith this circuit arrangement, practically all of the output noise voltage iE 250 K IN + 33V 5MUf Hf- 20on n-p-n 250 K 2.4 K 200 n 0.01/U.F p-n-p ■llOK 100 K POT. MANUAL ZERO SET I + 33V I + 4.5V OUT 5>UH -45V -27V +33V Fig. 8 — ■ DC summing amplifier. 310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 120 100 UJ U 80 LU o z " 60 < < IS LU a: tr D U Q. o o _) 40 20 ■20 -40 ,-' ../>' --T .-' — 364 LOCAL _ FEEDBACK"^- ^-' .-1 41,000 -^ ^ d 6 630 \, . 12 N \ ?,000 --.., s^-> ^-. 'S '-.. \ \ V \ \ V 2ND •^^ STAGE ^ s \ -.. "-. ^-. 1ST & 3RD STAGES N \ 0.5/ZF ^■-S;:-.-, ^ \, \ \ > \ 10' 10-^ 10- 10' 10' FRFOUENCY IN CYCLES PER SECOND Fig. 9 — Gain-frequency asymptotes for summing amplifier. is generated in the first transistor stage. If the transistor in the first stage has a noise figure less than 10 db at 1,000 cycles per second, then the RMS output noise voltage is less than 0.5 millivolts. Fig. 9 shows a plot of the gain-frequency asymptotes for the sum- ming amplifier determined from (13), (14), (15), (17), and (A6) under the assumption that the alphas and alpha-cutoff frequencies of the tran- sistors are 0.985 and 3 mc, respectively. The corner frequencies intro-' duced by the 0.5 microfarad condenser in the interstage network, thel local feedback circuit, and the cutoff of the first and third stages are so located that the current transmission falls off at an initial rate of about' 9 db per octave. This slope is joined to the final asymptote of the loop transmission by means of a step-type of transition.^ The transition is provided by 3 rising asymptotes due to the interstage shaping network, and the overall /S circuit. An especially large phase margin is used in order to insure a good transient performance. Fig. 10 shows the amplitude and phase of the loop current trans- mission. When the amplitude of the transmission is db, the phase angle is -292°, and when the phase angle is —360°, the amplitude is 27.5 db TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 311 100 LU m u LU Q <^ z < H Z UJ a. a. D o Q. O " o _l 80 60 40 20 20 -40 — ■~-^ ^•"v > \ ^ AM PLITL DE \ 1 \ \ \ s > \. ._ s >^ ^r •—'' y^ "N phase 'PHASE 'nCROSSOVER s \ \ V / GAIN^-N^ CROSSOVER N -27.5 DB 95 = -360° sv ■160 -200 to LU -240 ^ O LU Q -280 7 •320 -360 -400 ■440 10= FREQUENCY IN CYCLES PER SECOND 10^ 10' Fig. 10 — Loop current transmission of the summing amplifier. below db. The amplifier has a 68° phase margin and 27.5 db gain margin. In order to insure sufficient feedback at dc and adequate margins against instability, the transistors used in the amplifier should have alphas in the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater than 2.5 mc. 3.2. Automatic Zero Set of the dc Summing Amplifier The application of germanium junction transistors to dc amplifiers does not eliminate the problem of drift normally encountered in vacuum tube circuits. In fact, drift is more severe due principally to the varia- tion of the transistor parameters alpha and saturation current with temperature variation. Even though the amplifier has 80 db of negative feedback at dc, this feedback does not eliminate the drift introduced by [the first transistor stage. Because of the large amount of dc feedback, the collector current of the first stage is maintained relatively constant. The collector current of the transistor is related to the base current by the equation Ic = /c + a I — a 1 — a (18) [The saturation current, Ico , of a germanium junction transistor doubles (approximately for every 11°C increase in temperature. The factor a/(l — a) increases by as much as 6 db for a 25°C increase in tempera- 312 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ture. Consequently, the base current of the first stage, Ih , and the output voltage of the amplifier must change with temperature in order to main- ' tain Ic constant. The drift due to the temperature variation in a can be reduced by operating the first stage at a low value of collector current. With a germanium junction transistor in the first stage operating at a collector current of 0.25 milliamperes, the output voltage of the amplifier drifts about ±1.5 volts over a temperature range of 0°C to 50°C. It is possible to reduce the dc drift by using temperature sensitive elements in the amplifier. • In general, temperature compensation of a transistor dc amplifier requires careful selection of transistors and critical adjust- ment of the dc biases. However, even with the best adjustments, tem- perature compensation cannot reduce the drift in the amplifier to within typical limits such as ±5 millivolts throughout a temperature range of i to 50°C. In order to obtain the desired accuracy it is necessary to use an automatic zero set (AZS) circuit. t Fig. 11 shows a dc summing amplifier and a circuit arrangement fori reducing any dc drift that may appear at the output of the amplifier. The output voltage is equal to the negative of the sum of the input volt- ages, where each input voltage is multiplied by the ratio of the feedback resistor to its input resistor. In addition, an undesirable dc drift voltage ^ is also present in the ovitput voltage. The total output voltage is ^o.t = -i:^y|^ + Adrift (1!))^ In order to isolate the drift voltage, the A^ input voltages and the output voltage are applied to a resistance summing network composed of re- sistors Ro , Ri , R2 , • • • , Rn ■ The voltage across Rs is equal to Es=^ Adrift (20) if R,«Ro,R/; j = 1,2, ■■' ,N and RoRj = RkR,'; j = 1,2, ■■■ ,N The voltage E, is amplified in a relatively drift-free narrow band dc amplifier and is returned as a drift correcting voltage to the input of the dc summing amplifier. If the gain of the AZS circuit is large, the drift voltage at the output of the summing amplifier can be made very small. Fig. 12 shows the circuit diagram of a summing amplifier which uses a mechanical chopper in the AZS circuit.^^ The AZS circuit consists of a TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 313 resistance summing network, a 400-cycle synchronous chopper, and a tuned 400-cycle amplifier. Any drift in the summing amplifier will pro- duce a dc voltage Es at the output of the summing network. The chopper converts the dc voltage into a 400 cycles per second waveform. The fundamental frequency in the waveform is amplified by a factor of about 400,000 by the tuned amplifier. The synchronous chopper rectifies the sinusoidal output voltage and preserves the original dc polarity of Eg . The rectified voltage is filtered and fed back to the summing amplifier as an additional input current. The loop voltage gain of the AZS circuit at dc is about 54 db. Any dc or low-frequency drift in the summing amplifier is reduced by a factor of about 500 by the AZS circuit. The drift throughout a temperature range of to 50°C is reduced to ±3 millivolts. Since the drift in the summing amplifier changes at a relatively slow rate, the loop voltage gain of the AZS circuit can be cutoff at a relatively low frequency. In this particular case the loop voltage gain is zero db at about 10 cycles per second. 4.0. THE INTEGRATOR 4.1. Basic Design Considerations The design principles previously discussed are illustrated in this sec- tion by the design of a transistor integrator for application in a voltage VvV -OUT Fig. 11 — DC summing amplifier with automatic zero set. 314 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 315 encoder. The integrator is required to generate a 15-volt ramp which is linear and has a constant slope to within one part in 8,000. This ramp is to have a slope of 5 millivolts per microsecond for an interval of 3,000 microseconds. The first step in the design is to determine the bandwidth over which the negative feedback must be maintained in order to realize the desired output voltage linearity. The relationship between the output and input voltage of the integrator can be obtained from expression (11) by sub- stituting (1/pc) for Zk and R for Zj (refer to Fig. 1). £l-C'outJ — pRC A/3 + Zr^'pC 1 - AjS + -nN_ R (21) where ce[£'ouT] and JSiii'iN] are the Laplace transforms of the output and input voltages, respectively. In order to generate the voltage ramp, a step voltage of amplitude E is applied to the input of the integrator. The term Zy^ jR is negligible compared to unity at all frequencies. Therefore, £L-£'outJ — E \ A& + EZ IN 1 '^-RC Ll - A&\ pR \\ - A^_ It will be assumed that A/3 is given by the expression -K (22) A^ = V )0 + ^T (i + -M(i + ^ (23) Expression (23) implies that A/3 falls off at a rate of 6 db per octave at low frequencies and 12 db per octave at high frequencies. The output \ voltage of the integrator, as a function of time, is readily evaluated by substituting (23) into (22) and taking the inverse Laplace transform of the results. A good approximation for the output voltage is ^OUT — E RC + 2K ^-[(2w2+«l)(/2] ^;„ -x/W sm Vk> OJo ■iC02M ER (24)^ IN R [1 _ e-(-i'W _!_ g-[(2<-2+.i)t/2i ^Qg ^Tkc.,!] The linear voltage ramp is expressed by the term — (Et/RC) . The additional terms introduce nonlinearities. The voltage ramp has a slope of 5 millivolts per microsecond for E = —21 volts, R = 42,000 ohms, * In evaluating jE'out it was assumed that Zm' was equal to a fixed resistance Rin' , the low frequency input resistance to the first common emitter stage. A complete analysis indicates that this assumption makes the design conservative. 31G THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 and C = 0.1 microfarads. For these circuit values, and K = 10,000 (corresponding to 80 db of feedback) the nonhnear terms are less than 1/8,000 of the linear term (evaluated when / = 4 X 10"^ seconds) if /i ^ 30 cycles per second, J2 ^ 800 cycles per second, and if the first 1000 microseconds of the voltage ramp are not used. Consequently, 80 db of negative feedback must be maintained over a band extending from 30 to 800 cycles per second in order to realize the desired output voltage linearity. 4.2. Detailed Circuit Arrangement Fig. 13 shows the circuit diagram of the integrator. The method of biasing is the same as is used in the summing amplifier. The 200,000-ohm resistor provides approximately 0.5 milliamperes of collector current for the first stage. The 40,000-ohm resistor provides approximately 0.9 milliamperes of collector current for the second stage. The output stage is designed for a maximum power dissipation of 120 milliwatts and for an output voltage swing between —5 and +24 volts when operating into a load resistance equal to or greater than 40,000 ohms. J+'08V • + 108V 42 K D2 44- C 0.01>(/F o.l/iF 2.4K 270 K I + I08V 1MEG 200n \ — vw 2>U.F 200 K rVWA/^An j 100 K [ POT. I I OUT -10.5V + 108V + 4.5V •45V -10.5V Fig. 13 — Integrator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 317 1 !!3 { LU I 5 u ai a 1 z < 140 120 N 100 .^^ \ AMPLITUDE v. 80 ^""^ \ \ \ \ N \, 60 \ \ > \ \ \ s 40 20 \ '^— -- ,'' S "-s S, PHASE ^ \ \ ■\ PHASE . CROSSOVER -20 -40 GAIN-" CROSSOVER \ \ — ?n HR 95=- 360° ■80 -120 160 ■200 ■240 •280 UJ _J z < LU lO -320 < I Q. -360 -400 ■440 10 2 S .- 2 5 .^3 2 5 ,^^ 2 ^ 105 2 ^ ,0« ' ' 10^ lO'^ w FREQUENCY IN CYCLES PER SECOND Fig. 14 — - Loop current transmission of the integrator. The negative feedback in the integrator has been shaped by means of local feedback and interstage networks as described in Section 2.2. The loop current transmission has been calculated from (13), (14), (15), and (A6) and is plotted in Fig. 14. The transmission is determined under the assumption that the alphas of the transistors are 0.985 and the alpha- cutoff frequencies are three megacycles. Since the feedback above 800 cycles per second falls off at a rate of 9 db per octave, the analysis in Section 4.1 using (23), is conservative. The integrator has a 44° phase margin and a 20 db gain margin. In order to insure sufficient feedback between 30 and 800 cycles per second and adequate margins against instability, the transistors used in the integrator should have alphas in the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater than 2.5 megacycles. The silicon diodes Di and D2 are rec^uired in order to prevent the integrator from overloading. For output voltages between —4.0 and 21 volts the diodes are reverse biased and represent very high resistances, of the order of 10,000 megohms. If the output voltage does not lie in this range, then one of the diodes is forward biased and has a low resistance, of the order of 100 ohms. The integrator is then effectively a dc amplifier with a voltage gain of approximately 0.1. The silicon diodes affect the 318 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 linearity of the voltage ramp slightly due to their finite reverse resistances and variable shunt capacities. If the diodes have reverse resistances greater than 1000 megohms, and if the maximum shunt capacity of each diode is less than 10 micromicrofarads (capacity with minimum reverse voltage), then the diodes introduce negligible error. As stated earlier, the integrator generates a voltage ramp in response to a voltage step. This step is applied through a transistor switch which is actuated by a square wave generator capable of driving the transistor well into current saturation. Such a switch is required because the equivalent generator impedance of the applied step voltage must be very small. A suitable circuit arrangement is shown in Fig. 15. For the par- ticular application under discussion the switch *S is closed for 5,000 microseconds. During this time, the voltage E = —217 appears at the input of the integrator. At the end of this time interval, the transistor switch is opened and a reverse current is applied to the feedback con- denser C, returning the output voltage to —4.0 volts in about 2500 micro- seconds. An alternate way of specifying a low impedance switch is to say that the voltage across it be close to zero. For the transistor switch, con- nected as shown in Fig. 15, this means that its collector voltage be within FIRST STAGE OF DC AMPLIFIER 10.5V 50 K 150 K ' — WV-HVW RESIDUAL VOLTAGE BALANCE (TO AZS) Fig. 15 — Input circuit arrangement of the integrator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 319 one millivolt of ground potential during the time the transistor is in saturation. Xow, it has been shown that when a junction transistor in the common emitter connection is driven into current saturation, the minimum voltage between collector and emitter is theoretically equal to — in - (25) q oci where k is the Boltzmann constant, T is the absolute temperature, q is the charge of an electron ((kT/q) = 26 millivolts at room temperature), and ai is the inverse alpha of the transistor, i.e., the alpha with the emitter and collector interchanged. There is an additional voltage drop across the transistor due to the bulk resistance of the collector and emitter regions (including the ohmic contacts). A symmetrical alloy junction transistor with an alpha close to unity is an excellent switch because both the collector to emitter voltage and the collector and emit- ter resistances are very small. At the present time, a reasonable value for the residual voltage* be- tween the collector and emitter is 5 to 10 millivolts. This voltage can be eliminated by returning the emitter of the transistor switch to a small negative potential. This method of balancing is practical because the voltage between the collector and emitter of the transistor does not change by more than 1.0 millivolt over a temperature range of 0°C to 50°C. 4.3. Automatic Zero Set of the Integrator A serious problem associated with the transistor integrator is drift. The drift is introduced by two sources; variations in the base current of the first transistor stage and variations in the base to emitter potential of the first stage wdth temperature. In order to reduce the drift, the input resistor R and the feedback condenser C must be dissociated from the base current and base to emitter potential of the first transistor stage. This is accomplished by placing a blocking condenser Cb between point T and the base of the first transistor as shown in Fig. 15. An automatic zero set circuit is required to maintain the voltage at point T equal to zero volts. This AZS circuit uses a magnetic modulator known as a "magnettor."^^ A block diagram of the AZS circuit is shown in Fig. 16. The dc drift current at the input of the amplifier is applied to the magnettor. The carrier current required by the magnettor is supplied by a local transistor * The inverse alphas of the transistors used in this application were greater than 0.95. 320 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 oscillator. The useful output of the magnettor is the second harmonic of the carrier frequency. The amplitude of the second harmonic signal is proportional to the magnitude of the dc input current and the phase of the second harmonic signal is determined by the polarity of the dc input current. The output voltage of the magnettor is applied to an active filter which is tuned to the second harmonic frequency. The signal is then amplified in a tuned amplifier and applied to a diode gating circuit. Depending on the polarity of the dc input current, the gating circuit passes either the positive or negative half cycle of the second harmonic signal. In order to accomplish this, a square wave at a repetition rate equal to that of the second harmonic signal is derived from the carrier oscillator and actuates the gating circuit. A circuit diagram of the AZS circuit is shown in Figs. 17(a) and 17(b). The various sections of the circuit are identified with the blocks shown in Fig. 16. The active filter is adjusted for a Q of about 300, and the gain of the active filter and tuned amplifier is approximately 1000. The AZS circuit provides ±1.0 volt of dc output voltage for ±0.05 microamperes of dc input current. The maximum sensitivity of the circuit is limited to ±0.005 microamperes because of residual second harmonic generation in the magnettor with zero input current. When the transistor integrator is used together with the magnettor AZS circuit, the slope of the voltage ramp is maintained constant to within one part in 8,000 over a temperature range of 20°C to 40°C. 5.0. The Voltage Comparator The voltage comparator is one of the most important circuits used in analog to digital converters. The comparator indicates the exact time that an input waveform passes through a predetermined reference level. It has been common practice to use a vacuum tube blocking oscillator as a voltage comparator. ^^ Due to variations in the contact potential, heater voltage, and transconductance of the vacuum tube, the maximum DC INPUT AC MAGNETTOR ACTIVE FILTER \ GATING CIRCUIT ^ A ■~ OSCILLATOR GATING PULSE DC OUTPUT Fig. 16 — Block diagram of AZS circuit. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 321 accuracy of the circuit is limited to about ±100 millivolts. By taking advantage of the properties of semiconductor devices, the transistor blocking oscillator comparator can be designed to have an accuracy of ±5 millivolts throughout a temperature range of 20°C to 40°C. 5.1. General Descri'ption of the Voltage Comparator Fig. 18 shows a simplified circuit diagram of the voltage comparator. Except for the silicon junction diode D\ , this circuit is essentially a transistor blocking oscillator. For the purpose of analysis, assume that the reference voltage Vee is set equal to zero. When the input voltage V, is large and negative, the silicon diode Di is an open circuit and the jiuic- tion transistor has a collector current determined by Rb and Ebb [Expres- sion (18)]. The base of the transistor resides at approximately —0.2 volts. As the input voltage Vi approaches zero, the reverse bias across the diode Di decreases. At a critical value of Vi (a small positive poten- tial), the dynamic resistance of the diode is small enough to permit the circuit to become unstable. The positive feedback provided by trans- 1 former Ti forces the transistor to turn off rapidly, generating a sharp I output pulse across the secondary of transformer T-z . When Vi is large and positive, the diode Di is a low impedance and the transistor is main- tained cutoff. In order to prevent the comparator from generating more than one output pulse during the time that the circuit is unstable, the natural period of the circuit as a blocking oscillator must be properly chosen. Depending on this period, the input voltage waveform must have a certain minimum slope when passing through the reference level in order to prevent the circuit from misfiring. I The comparator has a high input impedance except during the switch- 1 ing interval.* When Vi is negative with respect to the reference level, the \ input impedance is equal to the impedance of the reverse biased silicon i diode. When Vi is positive with respect to the reference level, the input I impedance is equal to the impedance of the reverse biased emitter and ! collector junctions in parallel. This impedance is large if an alloy ; junction transistor is used. During the switching interval the input im- ■ pedance is equal to the impedance of a forward biased silicon diode in series with the input impedance of a common emitter stage (approxi- mately 1,000 ohms). This loading effect is not too serious since for the circuit described, the switching interval is less than 0.5 microseconds. The voltage comparator shown in Fig. 18 operates accurately on voltage waveforms with positive slopes. The voltage comparator will operate accurately on waveforms with negative slopes if the diode and * The switching interval is the time required for the transistor to turn off. 322 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 note: all capacitors and inductors IN tuned circuits have a tolerance of ±0.1% Fig. 17(a) — AZS circuit. battery potentials are reversed and if an n-p-n junction transistor is used. 5.2. Factors Determining the Accuracy of the Voltage Comparator Fig. 19 shows the ac equivalent circuit of the voltage comparator. In the equivalent circuit Ri is the dynamic resistance of the diode Di , Rg is the source resistance of the input voltage, and R2 is the impedance of TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 323 the load R^ as it appears at the primary of the transformer T2 . Ri is a function of the dc voltage across the diode Z)i . At a prescribed value of Ri , the comparator circuit becomes unstable and switches. The relation- ship between this critical value of Ri and the transistor and circuit parameters is obtained by evaluating the characteristic equation for the circuit and by determining the relationship which the coefficients of the equation must satisfy in order to have a root of the equation lie in the right hand half of the complex frequency plane. To a good approxima- tion, the critical value of Ri is given by the expression R, -\-R„ + n = Mao RiCc -\- (26) N'^Rr where M is the mutual inductance of transformer Ti and R2 — ly h^l Since the transistor parameters which appear in expression (26) have only a small variation with temperature, the critical value of Ri is independent of temperature (to a first approximation). It will now be shown that the comparator can be designed for an ac- curacy of ±5 millivolts throughout a temperature range of 20°C to 40°C. In order to establish this accuracy it will be assumed that the critical value of 7^1 is equal to 30,000 ohms. This assumption is based on the 30/iF TO LC FILTER IN MAGNETTOR NPUT CIRCUIT 4/iF +33V I+33V Fig. 17(b), 900-cycle carrier oscillator. 324 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 data displayed in Fig. 20 which gives the volt-ampere characteristics of a silicon diode measured at 20°C and 40°C. Throughout this temperature range, the diode voltage corresponding to the critical resistance of 30,000 ohms changes by about 30 millivolts. Fortunately, part of this voltage variation with temperature is compensated for by the variation in voltage Vb-e between the base and emitter of the junction transistor. From Fig. 18, V, = Vo - Vb-e + Ve (27) For perfect compensation (Vi independent of temperature), Vb-e should have the same temperature variation as the diode voltage Vd . Experi- REFERENCE I LEVEL -I ADJUSTMENT i+ Fig. 18 — Simplified circuit diagram of voltage comparator. Fig. 19 — Equivalent circuit of voltage comparator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 325 0.7 _) O > 0.6 R, = 30,000 OHMS 20°C > u:o.5 < I- _l o > o 0.3 2 3 4 5 6 DIODE CURRENT, Ip, IN MICROAMPERES Fig. 20 — Volt-ampere characteristic of a silicon junction diode. mentally it is found that Yh-e for germanium junction transistors varies by about 20 millivolts throughout the temperature range of 20°C to 40°C. Consequently, the variation in Yi at which the circuit switches is ±5 millivolts. It is apparent from Fig. 20 that the accuracy of the comparator in- creases slightly for critical values of R\ greater than 30,000 ohms, but decreases for smaller values. For example, the accuracy of the comparator is ±10 millivolts for a critical value of U\ equal to 5,000 ohms. In gen- eral, the critical value of R\ should be chosen between 5,000 and 100,000 ohms. 5.3. A Practical Yoltage Comparator Fig. 21 shows the complete circuit diagram of a voltage comparator. The circuit is designed to generate a sharp output pulse* when the input voltage waveform passes through the reference level (set by Yee) with a positive slope. The pulse is generated by the transistor switching from the "on" state to the "off" state. To a first approximation the amplitude of the output pulse is proportional to the transistor collector current during the "on" state. When the input voltage waveform passes through the reference level with a negative slope an undesirable negative pulse is generated. This pulse is eliminated by the point contact diode D2 . The voltage comparator is an unstable circuit and has the properties * For the circuit values shown in Fig. 21, the output pulse has a peak amplitude of about 6 volts, a rise time of 0.5 microseconds, and a pulse width of about 2.0 microseconds. 32G THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of a free running blocking oscillator after the input voltage Vi passes through the reference level. After a period of time the transistor will return to the "on" state unless the voltage Vi is sufficiently large at this time to prevent switching. In order to minimize the required slope of the hiput waveform the time interval between the instant Vi passes through the reference level and the instant the transistor would naturally switch to the "on" state must be maximized. This time intei-val can be con- trolled by connecting a diode D3 across the secondary winding of trans- former Ti . When the transistor turns off, the current which was flowing through the secondary of transformer Ti(Ic) continues to flow through the diode D3 so that L2 and D3 form an inductive discharge circuit. The point contact diode D3 has a forward dynamic resistance of less than 10 ohms and a forward voltage drop of 0.3 volt. If the small forward re- sistance of the diode is neglected, the time required for the current in the circuit to fall to zero is T = 0.3 (28) During the inductive transient, 0.3 volt is induced into the primary of transformer Ti (since N = 1) maintaining the transistor cutoff. The duration of the inductive transient can be made as long as desired by increasing L2 . However, there is the practical limitation that increasing L2 also increases the leakage inductance of transformer Ti , and in turn, I I -4.5V 5.1K 250A :iD2 >3K A-l- OUTPUT PULSE V- INPUT WAVEFORM PULSE AMPLITUDE^, ADJUSTMENT^ •^ 2.5 MEG POT. I- jr ee' Ij, = 4 MILS L, = L2= 5 MILLIHENRIES L', = L2= 5 MILLIHENRIES COEFFICIENT OF COUPLING = 0.99 REFERENCE g^ LEVEL ''adjustment MA 1 I I -46V I I -t-1.5V 100 OHM POT. I I -1.5V Fig. 21 — Voltage comparator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 327 increases the switching time. The circuit shown in Figure 21 does not misfire when used with voltage waveforms having slopes as small as 25 millivolts per microsecond, at the reference level. 6.0. A TRANSISTOR VOLTAGE ENCODER 6.1. Circuit Arrangement The transistor circuits previously described can be assembled into a voltage encoder for translating analog voltages into equivalent time intervals. This encoder is especially useful for converting analog informa- , tion (in the form of a dc potential) into the digital code for processing in a digital system. Fig. 22 shows a simplified block diagram of the encoder. The voltage I'amp generated by the integrator is applied to amplitude selector number one and to one input of a summing amplifier. The amplitude selector is a dc amplifier which amplifies the voltage ramp in the vicinity of zero volts. Voltage comparator number one, which follows the amplitude selector, generates a sharp output pulse at the exact instant of time that the voltage ramp passes through zero volts. The analog input voltage, which has a value between and —15 volts,* is applied to the second input of the summing amplifier. The output voltage of the summing amplifier is zero whenever the ramp INTEGRATOR N0.1 N0.1 3000^65 SUMMING AMPLIFIER AMPLITUDE SELECTORS VOLTAGE COMPARATORS ANALOG INPUT VOLTAGE 0-^-16V N0.2 N0.2 Fig. 22 — • Simplified block diagram of voltage encoder. * If the analog input voltage does not lie in this range, then the voltage gain of the summing amplifier must be set so that the analog voltage at the output of the summing amplifier lies in the voltage range between and +15 volts. 328 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 voltage is equal to the negative of the input analog voltage. At this instant of time the second voltage comparator generates a sharp output pulse. The time interval between the two output pulses is proportional to the analog input voltage if the voltage ramp is linear and has a con- stant slope at all times. 6.2. The Amplitude Selector i The amplitude selector increases the slope of the input voltage wave- form (in the vicinity of zero volts) sufficiently for proper operation of the voltage comparator. The amplitude selector consists of a limiter and a dc feedback amplifier as shown in Fig. 23. The two oppositely poled silicon diodes Di and D2 , limit the input voltage of the dc amplifier to about ±0.65 volts. The dc amplifier has a voltage gain of thirty, and so the maximum output voltage of the amplitude selector is limited to about ±19.5 volts. The net voltage gain between the input and output of the amplitude selector is ten. The principal requirement placed on the dc amplifier is that the input current and the output voltage be zero when the input voltage is zero. This is accomplished by placing a blocking condenser Cb between point T and the base of the first transistor stage, and by using an AZS circuit to maintain point T at zero volts. The dc and AZS amplifiers are identical in configuration to the amplifiers shown in Fig. 12. The dc amplifier is 50 K -VvV 50 K :|N D 1:: SILICON DIODES Dp 1.5 MEG Cb 250 /ZF 500 K I OUT I V^^ »— AAA^ 50 K 1.5 MEG -1 Fig. 23 — Block diagram of the amplitude selector. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 329 designed to have about 15.6 db less feedback than that shown in Fig. 10 since this amount is adequate for the present purpose. The bandwidth of the dc ampHfier is only of secondary importance because the phase shifts introduced by the two amplitude selectors in the voltage encoder tend to compensate each other. 6.3. Experimental Results The accuracy of the voltage encoder is determined by applying a precisely measured voltage to the input of the summing amplifier and by measuring the time interval between the two output pulses. The maxi- mum error due to nonlinearities in the summing amplifier and the voltage ramp is less than ±0.5 microseconds for a maximum encoding time of 3,000 microseconds. An additional error is introduced by the noise voltage generated in the first transistor stage of the summing amplifier. The ! RMS noise voltage at the output of the summing amplifier is less than 0.5 millivolts. This noise voltage produces an RMS jitter of 0.25 micro- I seconds in the position of the second voltage comparator output pulse. ; The over-all accuracy of the voltage encoder is one part in 4,000 through- ' out a temperature range of 20°C to 40°C. 1 I i ACKNOWLEDGEMENTS ! I The author wishes to express his appreciation to T. R. Finch for the ^ advice and encouragement received in the course of this work. D. W. ! Grant and W. B. Harris designed and constructed the magnettor used ' in the AZS circuit of the integrator. I Appendix I I RELATIONSHIP BETWEEN RETURN DIFFERENCE AND LOOP CURRENT i TRANSMISSION } In order to place the stability analysis of the transistor feedback ampli- fier on a sound basis, it is desirable to use the concept of return differ- ence. It will be shown that a measurable quantity, called the loop current transmission, can be related to the return difference of aZc with reference Ve .*• t In Fig. 24, N represents the complete transistor network exclusive of the transistor under consideration. The feedback loop is broken at the input to the transistor by connecting all of the feedback paths to * In this appendix it is assumed that the transistor under consideration is in the common emitter connection. The discussion can be readily extended to the other transistor connections. t This fact was pointed out by F. H. Tendick, Jr. 330 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Te+rb ^6^4 ( 'V -aZcLb N COMPLETE AMPLIFIER EXCLUSIVE OF THE TRANSISTOR IN QUESTION Fig. 24 — Measurement of loop current transmission. ground through a resistance (/•<; + n) and a voltage r J4 • Using the nomenclature given in Reference 8, the input of the complete circuit is designated as the first mesh and the output of the complete circuit is designated as the second mesh. The input and output meshes of the transistor under consideration are designated 3 and 4, respectively. The loop current transmission is equal to I3', the total returned current when a unit input current is applied to the base of the transistor. The return difference for reference Ve is equal to the algebraic differ- ence* between the unit input current and the returned current h'. 1 3 is evaluated by multiplying the open circuit voltage in mesh 4 (produced by the unit base current) by the backward transmission from mesh 4 to mesh 3 with zero forward transmission through the transistor under consideration. The open circuit voltage in mesh 4 is equal to (re — aZc). The backward transmission is determined with the element aZc , in the fourth row, third column of the circuit determinant, set equal to Ve . Hence, the return difference is expressed as A43 Fr' = 1 + {aZc - re) (Al)t Fr' = A''* + {aZc - r.)A 43 ir', (A2) Fr'.= A^'' = 1+ Tr' (A3) The relative return ratio Tr', is equal to the negative of the loop current transmission and can be measured as shown in Fig. 24. The voltage reh takes into account the fact that the junction transistor is not perfectly * The positive direction for the returned current is chosen so that if the original circuit is restored, the returned current flows in the same direction as the input current. t A''« is the network determinant with the element aZc in the fourth row, third column of the circuit determinant set equal to r, . TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 331 unilateral. Fortunately, in many applications, this voltage can be neg- lected even at the gain and phase crossover frequencies. In the case of single loop feedback amplifiers. A""* will not have any zeros in the right hand half of the complex frequency plane. A study of the stability of the amplifier can then be based on F^-, or T^-, . Appendix II INTERSTAGE NETWORK SHAPING This appendix presents the analysis of the circuit shown in Fig. 7(a). The input impedance of the common emitter connected junction tran- sistor is given by the expression ^iNPUT = n-\- re(l - Gl) (A4) where Gi is the current transmission of the common emitter stage, ex- pression (13). The current transmission A of the complete circuit is equal to A = ^ = ^ I\ Zz -\- ^ IN PUT G, (A5) where Z3 = i?3 + V^ + (l/p<^3). Combining (13), (A4), and (A5) yields ao A = 1 — ao + 5 1 + C03 + V \ W5/ I, Wl (A6) + p^ W5 , CsOO^in + Te -\- R3) _C0iC03- + PCO5 where WaWc(l — tto -}- 6) J ' CO3^C0aC0c(l — ^Q "j- 6) j ^ ^ Rl + Te COl = Wc = CO3 OJs = (1 - ao + 5) 1 + 6 _^ 1 1 (R^ + r,)Co 1 1 ~. . ^« C (1 - ao + 5)J^ 332 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Expression (A6) is valid if l/ws ^ 1/coi + RzCz . The denominator of the expression indicates a falUng 6 db per octave asymptote with a corner, frequency at ws . The second factor in the denominator can be approxi- mated bj^ a falHng 6 db per octave asymptote with a corner frequency at COl 1 ^^ n + (1 - ao + 5) ] n -\- Te -^ Rz -\- W1L3 pkis additional phase and amplitude contributions at higher f recjuencies due to the y and p terms. If COzCzRz then the circuit has a rising 12 db per octave asymptote with a corner frequency at C03 . Fig. 7(b) shows the amplitude and phase of the current transmission. REFERENCES 1. Felker, J. H., Regenerative Amplifier for Digital Computer Applications, Proc. I.R.E., pp. 1584-1596, Nov., 1952. 2. Korn, G. A., and Korn, T. M., Electronic Analog Computers, McGraw-Hil Book Company, pp. 9-19. 3. Wallace, R. L. and Pietenpol, W. J., Some Circuit Properties and Applications of n-p-n Transistors, B. S.T.J. , 30, pp. 530-563, July, 1951. 4. Shockley, W., Sparks, M. and Teal, G. K., The p-n Junction Transistor, Physical Review, 83, pp. 151-162, July, 1951. 5. Pritchard, R. L., Frequenc}' Variation of Current-Amplification for Junction Transistors, Proc. I.R.E., pp. 1476-1481, Nov., 1952. 6. Early, J. M., Design Theory of Junction Transistors, B.S.T.J., 32, pp. 1271- 1312, Nov., 1953. 7. Sziklai, G. C, Symmetrical Properties of Transistors and Their Applications, Proc. I.R.E., pp. 717-724, June, 1953. 8. Bode, H. W., Network Analysis and Feedback Amplifier Design, Van Nos- trand Co., Inc., Chapter IV. 9. Bode, H. W., Op. Cit., pp. 66-69. 10. Bode, H. W., Op. Cit., pp. 162-164. 11. Bargellini, P. M. and Herscher, M. B., Investigation of Noise in Audio Fre- quency Amplifiers Using Junction Transistors, Proc. I.R.E., pp. 217-226,' Feb., 1955. 12. Bode, H. W., Op. Cit., pp. 464-468, and pp. 471-473. 13. Keonjian, E., Temperature Compensated DC Transistor Amplifier, Proc: I.R.E., pp. 661-671, April, 1954. 14. Kretzmer, E. R., An Amplitude Stabilized Transistor Oscillator, Proc. I.R.E.,« pp. 391-401, Feb., 1954. i 15. Goldberg, E. A., Stabilization of Wide-Band Direct-Current Amplifiers for Zero and Gain, R.C.A. Review, June, 1950. 16. Ebers, J. J. and Moll, J. L., Large Signal Behavior of Junction Transistors. Proc. I.R.E., pp. 1761-1772, Dec, 1954. 17. Manlej', J. M., Some General Properties of Magnetic Amplifiers, Proc. I.R.K. March, 1951. 18. M.I.T., Waveforms, Volume 19 of the Radiation Laboratories Series. McGraw Hill Book Company, pp. 342-344. Electrolytic Shaping of Germanium , and Silicon ^ By A. UHLIR, JR. i (Manuscript received November 9, 1955) Properties of electrolyte-semiconductor barriers are described, with em- phasis on germanium. The use of these barriers in localizing electrolytic ! etching is discussed. Other localization techniques are mentioned. Electro- lytes for etching germanium and silicon are given. I INTRODUCTION I I Mechanical shaping techniques, such as abrasive cutting, leave the surface of a semiconductor in a damaged condition which adversely affects the electrical properties of p-n junctions in or near the damaged j material. Such damaged material may be removed by electrolytic etch- ing. Alternatively, all of the shaping may be done electrolytically, so that no damaged material is produced. Electrolytic shaping is particu- [ larly well suited to making devices with small dimensions. I A discussion of electrolytic etching can conveniently be divided into [■ two topics — the choice of electrolyte and the method of localizing the ji etching action to produce a desired shape. It is usually possible to find 1 an electrolyte in which the rate at which material is removed is accurately proportional to the current. For semiconductors, just as for metals, the I choice of electrolyte is a specific problem for each material ; satisfactory j electrolytes for germanium and silicon will be described. The principles of localization are the same, whatever the electrolyte used. Electrolytic etching takes place where current flows from the semiconductor to the electrolyte. Current flow may be concentrated at I certain areas of the semiconductor-electrolyte interface by controlling the flow of current in the electrolyte or in the semiconductor. LOCALIZATION IN ELECTROLYTE Localization techniques involving the electrolytic current are appli- cable to both metals and semiconductors. In some of these techniques, 333 334 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 the localization is so effective that the barrier effects found with n-type semiconductors can be ignored; if not, the barrier can be overcome by light or heat, as will be described below. If part of the work is coated with an insulating varnish, electrolytic etching will take place only on the uncoated surfaces. This technique, often called "masking," has the limitation that the etching undercuts the masking if any considerable amount of material is removed. The i same limitation applies to photoengraving, in which the insulating coat- ing is formed by the action of light. The cathode of the electrolytic cell may be limited in size and placed close to the work (which is the anode). Then the etching rate will be greatest at parts of the work that are nearest the cathode. Various shapes can be produced by moving the cathode with respect to the I work, or by using a shaped cathode. For example, a cathode in the form | of a wire has been used to slice germanium. Instead of a true metallic cathode, a "virtual cathode" may be used to localize electrolysis.^ In this technique, the anode and true cathode are separated from each other by a nonconducting partition, except for a small opening in the partition. As far as localization of current to the anode is concerned, the small opening acts like a cathode of equal size and so is called a virtual cathode. The nonconducting partition may include a glass tube drawn down to a tip as small as one micron diameter but nevertheless open to the flow of electrolytic current. With such a tip as a virtual cathode, micromachining can be conducted on a scale comparable to the wavelength of visible light. A general advantage of the virtual cathode technique is that the cathode reaction (usually hydrogen evolution) does not interfere with the localizing action nor with observation of the process. :| In the jet-etching technique, a jet of electrolyte impinges on the work.^'* The free streamlines that bound the flowing electrolyte are governed primarily by momentum and energy considerations. In turn, the shape of the electrolyte stream determines the localization of etch- ing. A stream of electrolyte guided by wires has been used to etch semi- conductor devices.^ Surface tension has an important influence on the free streamlines in this case, PROPERTIES OF ELECTROLYTE-SEMICONDUCTOR BARRIERS The most distinctive feature of electrolytic etching of semiconductors is the occurrence of rectifying barriers. Barrier effects for germanium will be described; those for silicon are qualitatively similar. The voltage-current curves for anodic n-type and p-type germanium ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 335 [in 10 per cent KOH are shown in Fig. 1. Tlie concentration of KOH [is not critical and other electrolytes give similar results. The voltage 'drop for the p-type specimen is small. For anodic n-type germanium, ! however, the barrier is in the reverse or blocking direction as evidenced by a large voltage drop. The fact that n-type germanium differs from p-type germanium only by very small amounts of impurities suggests that the barrier is a semiconductor phenomenon and not an electro- i chemical one. This is confirmed by the light sensitivity of the n-type 1 voltage-current characteristic. Fig. 2 is a schematic diagram of the ! arrangement for obtaining voltage-current curves. A mercury-mercuric loxide-10 per cent KOH reference electrode was used at first, but a gold (wire was found equally satisfactory. At zero current, a voltage Vo exists j between the germanium and the reference electrode ; this voltage is not [included in Fig. 1. I The saturation current Is , measured for the n-type barrier at a \moderate reverse voltage (see Fig. 1), is plotted as a function of tempera- Iture in Fig. 3. The saturation current increases about 9 per cent per [degree, just as for a germanium p-n junction, which indicates that the I 40 35 30 ^25 Lil O 20 15 10 1 12 OHM-CM n-TYPE / DAR\<. 1 / / 1 1 1 1 1 / 1 1 1 1 1 WITH ; LIGHT ^' 1 1 I 1 1 1 n i 1 1 / / P- FYPE 10 20 30 40 50 60 CURRENT FLOW IN MILLIAMPERES PER CM^ Fig. 1 — Anodic voltage-current characteristics of germanium. 336 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1956 current is proportional to the equilibrium density of minority carriers (holes). The same conclusion may be drawn from Fig. 4, which shows that the saturation current is higher, the higher the resistivity of the n-type germanium. But the breakdown voltages are variable and usu- ally much lower than one would expect for planar p-n junctions made, for example, by alloying indium into the same n-type germanium. Breakdown in bulk junctions is attributed to an avalanche multipli- cation of carriers in high fields.^ The same mechanism may be responsible for breakdown of the germanium-electrolyte barrier; low and variable breakdown voltages may be caused by the pits described below. The electrolyte-germanium barrier exhibits a kind of current multi- plication that differs from high-field multiplication in two respects: it occurs at much lower reverse voltages and does not vary much with voltage.^ This effect can be demonstrated very simply by comparison with a metal-germanium barrier, on the assumption that the latter has a current multiplication factor of unity. This assumption is supported by experiments which indicate that current flows almost entirely by hole flow, for good metal-germanium barriers. The experimental arrangement is indicated in Fig. 5(a) and (b). The voltage-current curves for an electrolyte barrier and a plated barrier on the same slice of germanium are shown in Fig. 5(c).* The curves for the REFERENCE ELECTRODE CATHODE LIGHT Fig. 2 — Arrangement for obtaining voltage current characteristics. * In Fig. 5 the dark current for the phited barrier is much hirger than can be exphained on the basis of hole current; it is even higher than the dark current for the electrolyte barrier, which should be at least 1.4 times the hole current. This excess dark current is believed to be leakage at the edges of the plated area and probably does not affect the intrinsic current multiplication of the plated barrier as a whole. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 337 10 2 o a. 01 a. to Ui oc LU Q. 5 < _) m cc tr 3 U z o cc 3 (0 •> I 10" / { - / - 1 / 7 / / / - /% - / / / / n/ ^ y i<:i 10 20 30 40 50 60 TEMPERATURE IN DEGREES CENTIGRADE Fig. 3 — Temperature variation of the saturation current of a barrier between 5.5 ohm-cm n-type germanium and 10 per cent KOH solution. illuminated condition were obtained by shining light on a dry face of a slice while the barriers were on the other face. The difference between the light and dark currents is larger for the electrolyte-germanium bar- rier than for the metal-germanium barrier, by a factor of about 1.4. The transport of holes through the slice is probably not very different for the two barriers. Therefore, a current multiplication of 1.4 is indi- cated for the electrolyte barrier. About the same value was found for temperatures from 15°C to 60°C, KOH concentrations from 0.01 per cent to 10 per cent, n-type resistivities of 0.2 ohm-cm to 6 ohm-cm, light currents of 0.1 to 1.0 ma/cm^, and for O.IN indium sulfate. Evidently the flow of holes to the electrolyte barrier is accompanied by a proportionate return flow of electrons, which constitutes an addi- tional electric current. Possible mechanisms for the creation of the electrons will be discussed in a forthcoming article. 338 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 7 > 4 LU o > I 0,5 1.0 1 CURRENT F 5 2.0 2.5 3.0 3.5 4.0 LOW IN MILLIAMPERES PER CM^ 4.5 Fig. 4. — Anodic voltage -current curves for various resistivities of germanium. SCRATCHES AND PITTING The voltage- current curve of an electrolyte-germanium barrier is very sensitive to scratches. The curves given in the illustrations were : obtained on material previously etched smooth in CP-4, a chemical I etch.* '' If, instead, one starts with a lapped piece of n-type germanium, the electrolyte-germanium barrier is essentially "ohmic;" that is, the voltage drop is small and proportional to the current. A considerable reverse voltage can be attained if lapped n-type germanium is electrolytically etched long enough to remove most of the damaged germanium. How- ever, a pitted surface results and the breakdown voltage achieved is not as high as for a smooth chemically-etched surface. The depth of damage introduced by typical abrasive sawing and lapping was investigated by noting the voltage-current curve of the Br2 Five parts HNO3 , 3 parts 48 per cent HF, 3 parts glacial acetic acid, ^0 P^-^t ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 339 electrolyte-germanium barrier after various amounts of material had been removed by chemical etching. After 20 to 50 microns had been re- moved, further chemical etching produced no change in the barrier characteristic. This amount of material had to be removed even if the lapping was followed by polishing to a mirror finish. The voltage-current curve of the electrolyte-germanium barrier will reveal localized damage. On the other hand, the photomagnetoelectric (PME) measurement of I -< — REFERENCE ELECTRODE CATHODE- -- -^ ■< ■y GLASS TUBING CEMENTED TO Ge E LECTROLYT z i N-Ge ■^ 1 1 1 1 <rri> (a) ELECTROPLATED INDIUM METAL TO N-Ge CONTACT ELECTROLYTE TO N-Ge BARRIER (c) 2 4 6 CURRENT, I, IN MILLIAMPERES PER CM 2 Fig. 5 — Determination of the current multiplication of the barrier between 6 ohm-cm n-type germanium and an electrolyte. 340 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Fig. 6 — Electrolytic etch pits on two sides of 0.02-inch slice of n-type germa- nium. Half of the slice was in contact with the electrolyte. surface recombination velocity gives an evaluation of the average con- dition of the surface. A variation of the PME method has been used to study the depth of abrasion damage; the damage revealed by this method extends only to a depth comparable to the abrasive size. A scratch is sufficient to start a pit that increases in size without limit if anodic etching is prolonged. However, a scratch is not necessary. Pits are formed even when one starts with a smooth surface produced by chemical etching. A drop in the breakdown voltage of the barrier is noticed when one or more pits form. The breakdown voltage can be restored by masking the pits with polystyrene cement. Evidence that the spontaneous pits are caused by some features of the crystal, itself, was obtained from an experiment on single-crystal n-type germanium made by an early version of the zone-leveling process. A slice of this material was electrolytically etched on both sides, after preliminary chemical etching. Photographs of the two sides of the slice are shown in Fig. 6. Only half of the slice was immersed in the electro- lyte. The electrolytic etch pits are concentrated in certain regions of the slice — the same general regions on both sides of the slice. It is interesting that radioautographs and resistivity measurements indicate high donor concentrations in these regions. Improvements, including more intensive stirring, were made in the zone-leveling process, and the electrolytic etch pit distribution and the donor radioautographs have been much more uniform for subsequent material. Several pits on a (100) face are shown in Fig. 7. The pits grow most rapidly in (100) directions and give the spiked effect seen in the illustra- tion. Toiler prolonged etching, the spikes and their branches form a com- plex network of caverns beneath the surface of the germanium. High-field carrier generation may be responsible for pitting. A locally ELECTROLYTIC SHAPING OF GERMAXIUM AND SILICON 341 Fig. 7 — Electrolytic etch pits on n-type germanium. high donor concentration would favor breakdown, as would any con- cavity of the germanium surface (which would cause a higher field for a given voltage) . Very high fields must occur at the points of spikes such jas those shown in Fig. 7. The continued growth of the spikes is thus favored by their geometry. Microscopic etch pits arising from chemical etching have been corre- ;lated with the edge dislocations of small-angle grain boundaries. A I specimen of n-type germanium with chemical etch pits was photomicro- graphed and then etched electrolytically. The etch pits produced elec- trolytically could not be correlated with the chemical etch pits, most of which were still visible and essentially unchanged in appearance. Also, no correlation could be found between either kind of etch pit and the locations at which copper crystallites formed upon immersion in a copper sulfate solution. Microscopic electrolytic etch pits at dislocations j in p-type germanium have been reported in a recent paper that also I mentions the deep pits produced on n-type germanium.^* y Electrolytic etch pits are observed on n-type and high-resistivity silicon. These etch pits are more nearly round than those produced in germanium. In spite of the pitting phenomenon, electrolytic etching is success- 342 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 I fully used in the fabrication of devices involving n-type semiconductors. Pitting can be reduced relative to "normal" uniform etching by any agency that increases the concentration of holes in the semiconductor. Thus, elevated temperatures, flooding with light, and injection of holes by an emitter all favor smooth etching. SHAPING BY MEANS OF INJECTED CARRIERS I Hole-electron pairs are produced when light is absorbed by semi- conductors. Light of short wavelength is absorbed in a short distance, while long wavelength light causes generation at considerable depths. The holes created by the light move by diffusion and drift and increase the current flow through an anodic electrolyte-germanium barrier at whatever point they happen to encounter the barrier. In general, more holes will diffuse to a barrier, the nearer the barrier is to the point at which the holes are created. For n-type semiconductors, the current due to the light can be orders of magnitude greater than the dark cur- rent, so that the shape resulting from etching is almost entirely deter- mined by the light. As shown in Fig. 3, the dark current can be made very small by lowering the temperature. An example of the shaping that can be done with light is shown in Fig. 8. A spot of light impinges on one side of a wafer of n-type germanium or silicon. The semiconductor is made anodic with respect to an etching electrolyte. Accurately concentric dimples are produced on both sides of the wafer. Two mechanisms operate to transmit the effect to the oppo- site side. One is that some of the light may penetrate deeply before generating a hole-electron pair. The other is that a fraction of the car- riers generated near the first surface will diffuse to the opposite side. By varying the spectral content of the light and the depth within the \ \ -n-TYPE SEMICONDUCTOR LIGHT I I Fig. 8 — Double dimpling with light. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 343 wafer at which the light is focused, one can produce dimples with a vari- ,'ety of shapes and relative sizes. I It is obvious that the double-dimpled wafer of Fig. 8 is desirable for {the production of p-n-p alloy transistors. For such use, one of the most [important dimensions is the thickness remaining between the bottoms of the two dimples. As has been mentioned in connection with the jet- I etching process, a convenient way of monitoring this thickness to de- Itermine the endpoint of etching is to note the transmission of light of [suitable wavelength.^ There is, however, a control method that is itself [automatic. It is based on the fact that at a reverse-biased p-n junction [Or electrolyte-semiconductor barrier there is a space-charge region that is practically free of carriers. When the specimen thickness is reduced so that space-charge regions extend clear through it, current ceases to flow and etching stops in the thin regions, as long as thermally or op- tically generated carriers can be neglected. However, more pitting is to be expected in this method than when etching is conducted in the pres- ence of an excess of injected carriers. A p-n junction is a means of injecting holes into n-type semiconduc- tors and is the basis of another method of dimpling, shown in Fig. 9. The p-n junction can be made by an alloying process such as bonding an acceptor-doped gold wire to germanium. The ohmic contact can be made by bonding a donor-doped gold wire and permits the injection of a greater excess of holes than would be possible if the current through the p-n junction were exactly equal to the etching current. Dimpling without the ohmic contact has been reported.^ 14 OHMIC CONTACT p-n JUNCTION Fig. 9 — Dimpling with carriers injected by a p-n junction. 344 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CONTROL BY OHMIC CONDUCTION The carrier-injection shaping techniques work very well for n-typei material. It is also possible to inject a significant number of holes intos rather high resistivity p-type material. But what can be done about: p-type material in general, short of developing cathodic etches? ] The ohmic resistivity of p-type material can be used as shown in Fig.!^ 10. More etching currect flows through surfaces near the small contact than through more remote surfaces. A substantial dimpling effect is observed when the semiconductor resistivity is equal to the electrolyte resistivity, but improved dimpling is obtained on higher resistivity semiconductor. This result is just what one might expect. But the math- ematical solution for ohmic flow from a point source some distance from a planar boundary between semi-infinite materials of different conduc- tivities shows that the current density distribution does not depend on the conductivities. An important factor omitted in the mathematical solution is the small but significant barrier voltage, consisting largely of electrochemical polarization in the electrolyte. The barrier voltage is; approximately proportional to the logarithm of the current density; while the ohmic voltage drops are proportional to current density. Thus,- high current favors localization. ELECTROLYTES FOR ETCHING GERMANIUM AND SILICON » The electrolyte usually has two functions in the electrolytic etching of an oxidizable substance. First, it must conduct the current necessary for the oxidation. Second, it must somehow effect removal of the oxida- tion product from the surface of the material being etched. The usefulness of an electrolytic etch depends upon one or both of: ANY CONTACT, PREFERABLY OHMIC ^//yyyy//y/y/y////////y////y///// yyyyyyyyyyy7^ Fig. 10 — Dimpling by ohmic conduction. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 345 the following situations — the electrolytic process accomplishes a reac- tion that cannot be achieved as conveniently in any other way or it permits greater control to be exercised over the reaction. Accordingly, chemical attack by the chosen electrolyte must be slight relative to the electrochemical etching. A smooth surface is probably desirable in the neighborhood of a p-n junction, to avoid field concentrations and lowering of breakdown voltage. Therefore, a tentative requirement for an electrolyte is the production of a smooth, shiny surface on the p-type semiconductor. Such \ an electrolyte will give a shiny but possibly pitted surface on n-type j specimens of the same semiconductor. The effective valence of a material being electrolytically etched is ; defined as the number of electrons that traverse the circuit divided by the number of atoms of material removed. (The amount of material ! removed was determined by weighing in the experiments to be described.) If the effective valence turns out to be less than the valence one might predict from the chemistry of stable compounds, the etching is sometimes said to be "more than 100 per cent efficient." Since the anode reactions in electrolytic etching may involve unstable intermediate compounds and competing reactions, one need not be surprised at low or fractional effective valences. Germanium can be etched in many aqueous electrolytes. A valence of almost exactly 4 is found. That is, 4 electrons flow through the circuit for each atom of germanium removed. For accurate valence measure- ments, it is advisable to exclude oxygen by using a nitrogen atmosphere. Potassium hydroxide, indium sulfate, and sodium chloride solutions are among those that have been used. Sulfuric acid solutions are prone to ) yield an orange-red deposit which may be a suboxide of germanium/* I Similar orange deposits are infrequently encountered with potassium I hydroxide. Hydrochloric acid solutions are satisfactoiy electrolytes. The reaction I product is removed in an unusual manner when the electrolyte is about 2N hydrochloric acid. Small droplets of a clear liquid fall from the etched regions. These droplets may be germanium tetrachloride, which is denser than the electrolyte. They turn brown after a few seconds, perhaps be- cause of hydrolysis of the tetrachloride. Etching of germanium in sixteen different aqueous electroplating electrolytes has been mentioned. Germanium can also be etched in the partly organic electrolytes described below for silicon. One would expect that silicon could be etched by making it the anode in a cell with an aqueous hydrofluoric acid electrolyte. The seemingly 346 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 | ) likely oxidation product, silicon dioxide, should react with the hydro-! fluoric acid to give silicon tetrafluoride, which could escape as a gas. In fact, a gas is formed at the anode and the silicon loses weight. But the gas is hydrogen and an effective valence of 2.0 ± 0.2 (individual deter- minations ranged from 1.3 to 2.7) was found instead of the value 4 that i might have been expected. The quantity of hydrogen evolved is con- sistent with the formal reaction Si —> Si"*"'" + me (electrochemical oxidation) Si+™ + (4-to)H+ -^ Si+' + Vz (4-m)H2 (chemical oxidation) where m is about two. The experiments were done in 24 per cent to 48 per cent aqueous solutions of HF at current densities up to 0.5 amp/cm^. The suggestion that the electrochemical oxidation precedes the chemi- cal oxidation is supported by the appearance and behavior of the etched surfaces. Instead of being shiny, the surfaces have a matte black, brown, or red deposit. At 40 X magnification, the deposit appears to consist of flakes of a; resinous material, tentatively supposed to be a silicon suboxide. A re- markable reaction can be demonstrated if the silicon is rinsed briefly in water and alcohol after the electrolytic etch, dried, and stored in air for as long as a year. Upon reimmersing this silicon in water, one can observe the liberation of gas bubbles at its surface. This gas is presumed to be hydrogen. To initiate the reaction it is sometimes necessary to dip the specimen first in alcohol, as water may otherwise not wet it. The speci- mens also liberate hydrogen from alcohol and even from toluene. Thus, chemical oxidation can follow electrolytic oxidation. But chemical oxidation does not proceed at a significant rate before thei current is turned on. Smooth, shiny electrolytic etching of p-type silicon has been obtained; with mixtures of hydrofluoric acid and organic hydroxyl compounds,; such as alcohols, glycols, and glycerine. These mixtures may be an- hydrous or may contain as much as 90 per cent water. The organic additives tend to minimize the chemical oxidation of the silicon. They; also permit etching at temperatures below the freezing point of aqueous solutions. They lower the conductivity of the electrolyte. For a given electrolyte composition, there is a threshold current density, usually between 0.01 and 0.1 amps/cm , for smooth etching.; Lower current densities give black or red surfaces with the same hy- drogen-liberating capabilities as those obtained in aqueous hydrofluoric acid. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 347 In general, smooth etching of siHcon seems to result when the effective valence is nearly 4 and there is little anodic evolution of gas. The elec- I trical properties of the smooth surface appear to be equivalent to those ! of smooth silicon surfaces produced by chemical etching in mixtures of i nitric and hydrofluoric acids. On the other hand, the reactive surface [produced at a valence of about 2, with anodic hydrogen evolution, is I capable of practically shorting-out a silicon p-n junction. The electrical j properties of this surface tend to change upon standing in air. ACKNOWLEDGEMENTS Most of the experiments mentioned in this paper were carried out by my wife, Ingeborg. An exception is the double-dimpling of germanium by light, which was done by T. C. Hall. The dimpling procedures of Figs. 9 and 10 are based on suggestions by J. M. Early. The effect of light upon electrolytic etching was called to my attention by 0. Loosme. W. G. Pfann provided the germanium crystals grown with different degrees of stirring. REFERENCES 1. J. F. Barry, I.R.E.-A.I.E.E. Semiconductor Device Research Conference, Philadelphia, June, 1955. 2. A. Uhlir, Jr., Rev. Sci. Inst., 26, pp. 965-968, 1955. 3. W. E. Bailey, U. S. Patent No. 1,416, 929, May 23, 1922. 4. Bradley, et al. Proc. I.R.E., 24, pp. 1702-1720, 1953. 5. M. V. Sullivan and J. H. Eigler, to be published. 6. S. L. Miller, Phys. Rev. 99, p. 1234, 1955. 7. W. H. Brattain and C. G. B. Garrett, B.S.T.J., 34, pp. 129-176, 1955. 8. E. H. Borneman, R. F. Schwarz, and J. J. Stickler, J. Appl. Phvs., 26, pp. 1021-1029, 1955. 9. D. R. Turner, to be submitted to the Journal of the Electrochemical Society. 10. R. D. Heidenreich, U. S. Patent No. 2,619,414, Nov. 25, 1952. 11. T. S. Moss, L. Pincherle, A. M. Woodward, Proc. Phys. Soc. London, 66B, p. 743, 1953. 12. T. M. Buck and F. S. McKim, Cincinnati Meeting of the Electrochemical Society, Mav, 1955. 13. F. L. Vogel, W. G. Pfann, H. E. Corey, and E. E. Thomas, Phys. Rev., 90, p. 489, 1953. 14. S. G. Ellis, Phys. Rev., 100, pp. 1140-1141, 1955. 15. Electronics, 27, No. 5, p. 194, May, 1954. 16. F. Jirsa, Z. f. Anorg. u. AUgemeine Chem., Bd. 268, p. 84, 1952. \ A Large Signal Theory of Traveling Wave Amplifiers Including the Effects of Space Charge and Finite Coupling Between the Beam and the Circuit By PING KING TIEN Manuscript received October 11, 1955) The non-linear behavior of the traveling-wave amplifier is calculated in this paper by numericalhj integrating the motion of the electrons in the presence of the circuit and the space charge fields. The calculation extends the earlier work by Nordsieck and the srnall-C theory by Tien, Walker and Wolontis, to include the space charge repulsion between the electrons and the effect of a finite coupling between the circuit and the electron beam. It however differs from Poulter's and Rowers works in the methods of calcu- lating the space charge and the effect of the backward wave. The numerical work was done using 701 -type I.B.M. equipment. Re- sults of calcidation covering QC from 0.1 to 0.4, b from 0.46 to 2.56 and k from 1.25 to 2.50, indicate that the saturation efficiency varies between 23 per cent and 37 per cent for C equal to 0.1 and between 33 per cent and Jf.0 per cent for C equal to 0.15. The voltage and the phase of the circuit wave, the velocity spread of the electrons and the fundamental component of the charge-density modidation are either tabulated or presented in curves. A method of calculating the backward wave is provided and its effect fully discussed. 1. INTRODUCTION Theoretical evaluation of the maximum efficiency attainable in a traveling-wave amplifier requires an understanding of the non-linear behavior of the device at various working conditions. The problem has been approached in many ways. Pierce/ and later Hess,^ and Birdsalf and Caldwell investigated the efficiency or the output power, using cer- tain specific assumptions about the highly bunched electron beam. They either assume a beam in the form of short pulses of electrons, or, specify 349 350 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 an optimum ratio of the fundamental component of convection current to the average or d-c current. The method, although an abstract one, generally gives the right order of the magnitude. When the usual wave concept fails for a beam in which overtaking of the electrons arises, we may either overlook effects from overtaking, or, using the Boltzman's transport equation search for solutions in series form. This attack has been pursued by Parzen and Kiel, although their work is far from com- plete. The most satisfying approach to date is Nordsieck's analysis.' Nordsieck followed a typical set of "electrons" and calculated their velocities and positions by numerically integrating a set of equations of motion. Poulter has extended Nordsieck equations to include space charge, finite C and circuit loss, although he has not perfectly taken into account the space charge and the backward wave. Recently Tien, Walker, and Wolontis have published a small C theory in which "elec- trons" are considered in the form of uniformly charged discs and the space charge field is calculated by computing the force exerted on one disc by the others. Results extended to finite C, have been reported by Rowe,^*^ and also by Tien and Walker.^^ Rowe, using a space charge expression similar to Poulter's, computed the space charge field based on the electron distribution in time instead of the distribution in space. This may lead to appreciable error in his space charge term, although its influence on the final results cannot be easily evaluated. In the present analysis, we shall adopt the model described by Tien, Walker and Wolontis, but wish to add to it the effect of a finite beam to circuit coupling. A space charge expression is derived taking into account the fact that the a-c velocities of the electrons are no longer small com- pared with the average velocity. Equations are rewritten to retain terms involving C. As the backward wave becomes appreciable when C in- creases, a method of calculating the backward wave is provided and the effect of the backward wave is studied. Finally, results of the calculation covering useful ranges of design and operating parameters are presented and analyzed. 2. ASSUMPTIONS To recapitulate, the major assumptions which we have made are: 1. The problem is considered to be one dimensional, in the sense that the transverse motions of the electrons are prohibited, and the current, velocity, and fields, are functions only of the distance along the tube and of the time. 2. Only the fundamental component of the current excites waves on the circuit. A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 351 3. The space charge field is computed from a model in which the helix is replaced by a conducting cylinder, and electrons are uniformly charged discs. The discs are infinitely thin, concentric with the helix and have a radius equal to the beam radius. 4. The circuit is lossfree. These are just the assumptions of the Tien-Walker-Wolontis model. In addition, we shall assume a small signal applied at the input end of a long tube, where the beam entered unmodulated. What we are looking for are therefore the characteristics of the tube beyond the point at which the device begins to act non-linearly. Let us imagine a flow of electron discs. The motions of the discs are computed from the circuit and the space charge fields by the familiar Newton's force equation. The elec- trons, in turn, excite waves on the circuit according to the circuit equa- tion derived either from Brillouin's model^ or from Pierce's equivalent circuit. The force equation, the circuit equation, and the equation of conservation of charge in kinematics, are the three basic equations from which the theory is derived. 3. FORWARD AND BACKWARD WAVES In the traveling-wave amplifier, the beam excites forward and back- ward waves on the circuit. (We mean by "forward" wave, the wave which propagates in the direction of the electron flow, and by "back- ward" wave, the wave which propagates in the opposite direction.) Because of phase cancellation, the energy associated with the backward wave is small, but increases with the beam to circuit coupling. It is there- fore important to compute it accurately. In the first place, the waves on the circuit must satisfy the circuit equation dH^(z,t) 2d'V{z,t) „ d'p^iz, t) ,v Here, V is the total voltage of the waves. Vo and Zo are respectively the phase velocity and the impedance of the cold circuit, z is the distance along the tube and t, the time, p^ is the fundamental component of the linear charge density. V and p„ are functions of z and /. The complete solution of (1) is in the form Viz) = Cre'^'' + (726 "^"^ + e —-y— J e " po,{^) dz ^2) + e " —^ j e p^{z) dz 352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 where the common factor e^"' is omitted. To = j{co/vo), j = \/— 1 and w is the angular frequency. Ci and C2 are arbitrary constants which will be determined by the boundary conditions at the both ends of the beam. The first two terms are the solutions of the homogeneous equation (or the complementary functions) and are just the cold circuit waves. The third and the fourth terms are functions of electron charge density and are the particular solution of the equation. Let us consider a long traveling-wave tube in which the beam starts from z = and ends at 2; = D. The motion of electrons observed at any particular position is periodic in time, though it varies from point to point along the beam. To simplify the picture, we may divide the beam along the tube into small sections and consider each of them as a current element uniform in z and periodic in time. Each section of beam, or each current element excites on the circuit a pair of waves equal in ampli- tudes, one propagating toward the right (i.e., forward) and the other, toward the left. One may in fact imagine that these are trains of waves supported by the periodic motion of the electrons in that section of the beam. Obviously, a superposition of these waves excited by the whole beam gives the actual electromagnetic field distribution on the circuit. One may thus compute the forward traveling wave at z by summing all the waves at z which come from the left. Stated more specifically, the forward traveling energy at z is contributed by the waves excited by the current elements at the left of the point z. Similarly the backward travel- ing energy, (or the backward wave) at z is contributed by the waves excited by the current elements at the right of the point z. It follows obviously from this picture that there is no forward wave at 2 = (except one corresponding to the input signal), and no backward wave at 2 = D. (This implies that the output circuit is matched.) With these boundary conditions, (1) is reduced to z) = Finput e " + e ° — -— / e " po,{z) Z Jo dz + /-^J e-%.(.) (3) dz Equations (2) and (3) have been obtained by Poulter.^ The first term of (3) is the wave induced by the input signal. It propagates as though the ; beam were not present. The second term is the voltage at z contributed by the charges between 2 = and 2 = 2. It is just the voltage of the forward wave described earlier. Similarly the third term which is the voltage at 2 contributed by the charges between z = z and 2 = D is the voltage of the backward wave at the point 2. Denote F and B respec- A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 353 tively the voltages of the forward and the backward waves, we have F{z) = Fi„put e-'"^ + e-^»^ ^« r e'^' p^z) dz (4) Z Jo Biz) = e^- ^° £ e-^-p„(e) dz (5) It can be shown by direct substitution that F and B satisfy respectively the differential equations dz Vo dt 2 (9^ (6) dB(z, t) 1 a5(2, Zo ap„(2, (92 1^0 di 2 dt (7) We put (4) and (5) in the form of (6) and (7) simply because the differ- ential equations are easier to manipulate than the integral equations. In fact, we should start the analysis from (6) and (7) if it were not for a physical picture useful to the understanding of the problem. Equations (6) and (7) have the advantage of not being restricted by the boundary conditions at 2; = and D, which we have just imposed to derive (4) and (5). Actually, we can derive (6) and (7) directly from the Brillouin model in the following manner. Suppose Y, I and Zo are respectively the voltage, current and the characteristic impedance of a transmission line system in the usual sense. (V + /Zo) must then be the forward wave and {V — IZo) must be the backward wave. If we substituted F and B in these forms into (1) of the Brillouin' s paper,^^ we should obtain exactly (6) and (7). It is obvious that the first and third terms of (2) are respectively the complementary function and the particular solution of (6), and similarly the second and the fourth terms of (2) are respectively the comple- mentary function and the particular solution of (7). From now on, we shall overlook the complementary functions which are far from syn- chronism with the beam and are only useful in matching the boundary conditions. It is the particular solutions which act directly on the elec- tron motion. With these in mind, it is convenient to put F and B in the form Fiz, t) = -j~ [aiiij) cos <p - aiiy) sin ^] (8) B{z, t) = -^ [hiiy) cos ip - h^iy) sin 9?] (9) 354 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 where ai(y), 02(2/), hi(y) and 62(2/) are functions of y. y and <p are inde- pendent variables and have been used by Nordsieck to replace the vari- ables, z and t, such as y = C — Z (f = w [ — — t ] \Vo / Here as defined earlier, I'o is the phase velocity of the cold circuit and Vq the average velocity of the electrons. They are related by the parameter h defined by Pierce as Uo 1 vo (1 - hC) C is the gain parameter also defined by Pierce, ^3 _ ZqIo in which, Vo and 7o are respectively the beam voltage and current. Adding (6) to (7), we obtain an important relation between F and B, that is, dFjz, t) _^ 1_ dF{z, t) ^ dBjz, t) _j_ l_ dBjz, t) ^^Q^ dz Vo dt dz Vo dt Substituting (8) and (9) into (10) and carrying out some algebraic manipulation, we obtain '"'^^ = " 2(1 + bC) I ^'^^'> + "'-^"^^ (11) "'^^'^ = 2(1 + bC) ly '"'^^^ + '"^^^' or B{z, t) = ZqIo C dMy) + bM) ,„, ^ + diaM+ b.(,j)) ^.^ - dy dy [ For better understanding of the problem, we shall first solve (12a) ap- proximately. Assuming for the moment that hiiy) and h^^y) are small compared with ai{ij) and a^iy) and may be neglected in the right-hand A LARGE SIGNAL THEORY OP TRAVELING-WAVE AMPLIFIERS 355 member of the equation, we obtain for the first order solution iKz, t) ^ ZqIo I (^ sin <p + — ^^^ cos <p 40 \ 2(1 + bC) I dij ^ ' dy (12b) Of course, the solution (12b) is justified only when hi(y) and ?)2(?y) thus obtained are small compared with ai(y) and aoiy). The exact solution of B obtained by successive approximation reads Biz, t) + ZqIo I c 4(7 V 2(1 + bC) 4(1 + hC) It may be seen that the term involving dai(y) . , da2(ij) -^ sm <p + , cos _ dy dy ■ ] •] '^^' cos<p + — f^sm^ + (12c) dy- dy- 4(1 + bcy and the higher order terms are neglected in our approximate solution. For C equal to few tenths, the difference between (r2b) and (12c) only amounts to few per cent. We thus can calculate the backward wave by (12b) or (12c) from the derivatives of the forward wave. To obtain the complete solution of the backward wave, we should add to (12b) or (12c) a solution of the homogeneous equation. We shall return to this point later. 4. WORKING EQUATIONS With this discussion of the backward wave, we are now in a position to derive the working equations on which our calculations are based. In Nordsieck's notation, each electron is identified by its initial phase. Thus, (p(y, (fo) and Cuow(y, <po) are respectively the phase and the ac velocity of the electron which has an initial phase (fo . It should be remem- bered that y is equal to and is used by Nordsieck as an independent variable to replace the vari- al)le z. Let us consider an electron which is at Zo when /, = and is at z (or ?/) when t = /. Its initial phase is then OiZo <Po = — 356 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 and its phase at y is <p(y,<po) = oj f- - tj i The velocity of the electron is expressed as dz dt = Wo[l + Cw{ij, ip^)] where Uo is the average velocity of the electrons, and, Cuow(y, tpo) as men- tioned earlier, is the ac velocity of the electron when it is at the position y. The electron charge density near an electron which has an initial phase cpo and which is now at y, can be computed by the equation of conserva- tion of charge, it is p(y, <Po) = - Wo d(po d(p{y, <po) 1 1 + Cw(y, ifo) (13) One should recall here that h is the dc beam current and has been de- fined as a positive quantity. When several electrons with different initial phases are present at y simultaneously, a summation of d<po of these electrons should be used in (13). From (13), the fundamental component of the electron charge density is pMt) = --- sm d<po sin (fiy, <po) 1 + Cw{y, <pq) r^" , cos <p{y, <po) + cos <p I d(po Jo (14) 1 + Cw(y, ifo)/ These are important relations given by Nordsieck and should be kept in mind in connection with later work. In addition, we shall frequently use the transformation I = t s = ^"(' + ^'"(^-» 1^ which is written following the motion of the electron. Let us start from the forward wave. It is computed by means of (6). After substituting (8) and (14) into (6), we obtain by equating the sin <p and the cos v' A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 357 terms dax{y) ^ _2 T^" ^ sin <p(y, cpo) .^. dy IT h " 1 + Cwiv, (po) da.Xy) 2 f^" cos<p{y,<po) ..^n — 1 — = ~- / d(po , r (.lb; dy IT Jo 1 + Cw{y, <po) Next we shall calculate the electron motion. We express the acceleration of an electron in the form d'z „ /I I /o / ^^ dw{y, <po) ^ = Cuod + Cw{y, M -^^ and calculate the circuit field by differentiating F in (8) and B in (12c) with respect to z. One thus obtains from Newton's law 2[1 + Cw{y, <po)] ^^'^J' ^°^ = (1 + hOMy) sin <p + a,{y) cos <p\ dy + ^-^ r^ «in ^ + ^^ cos J - -^ ^. 4(1 + 6C) L ^Z/- c?^^ J WomwC^ Here Eg is the space charge field, which will be discussed in detail later. Finally a relation between w{y, (po) and <p{y, ^o) is obtained by means of (13) difiy, <po) _ ^ ^ ^^(y, <Po) QgN dy 1 + Cw{y, <pq) Equations (15), (16), (17) and (18) are the four working equations which we have derived for finite C and including space charge. Instead of writing the equations in the above form, Rowe, ignoring the backward wave, derives (15) and (16) directly from the circuit equation (1). He obtains an additional term C d^tti 2 dy"" for (15) and another term C d"ai 2df for (16). It is apparent that the backward wave, though generally a small quantity, does influence the terms involving C. 358 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 5. THE SPACE CHARGE EXPRESSION We have mentioned earlier that the space charge field is computed from the disc-model suggested by Tien, Walker and Wolontis. In their calculation, the force excited on one disc by the other is approximated by an exponential function F. = — [a(z'— z)/ro] 27rro-eo Here ro is the radius of the disc or the beam, q is the charge carried by each disc, and eo is the dielectric constant of the medium. The discs are supposed to be respectively at z and z . a is a constant and is taken equal to 2. Consider two electrons which have their initial phases <pq and ^o and which reach the position ij (or z) at times t and t' respectively. The time difference, * - / = 1 00 wt — — Z — [bit — — Z) Vo \ Vq J CO multiplied by the velocity of the electron i<o[l + Cw(y, (po )] is obviously the distance between the two electrons at the time t. Thus (z - z)t=t = - y(y, <Po) - <p(y, <Po)]uo[l + Cw(y,ipo)] (19a) In this equation, we are actually taking the first term of the Taylor's expansion, (z — z)t=t = dzjij, cpo) dt t=t (t _ /^ j_ ^ c?^2(y, <pq) it - ty t=t (19b) + It is clear that the electrons at y may have widely different velocities after having traveled a long distance from the input end, but changes in their velocities, in the vicinity of y and in a time-period of around 2 tt, are relatively small. This is why we must keep the first term of (19b) and may neglect the higher order terms. From (19a) the space charge field Es in (17) is 2e Es = /+00 -k]ip(.y ,<po+<t>)—<p(.U ,<Po) 1 li+Cw(y,(po+<t>)] d(f> sgn (<p(<po -\- <p) - <t>iy, <po)) Here, e/m is the ratio of electron charge to mass, cop is the electron A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 359 angular plasma frequency for a beam of infinite extent, and k is 2 k = a 0) CO — ro — ro Uo Wo (20) In the small C theory, th^e distribution of electrons in time or in time- phase at z is approximately the same as the distribution in z (also ex- pressed in the unit of time-phase) at the vicinity of z. This is, however, not true when C becomes finite. The difference between the time and space distributions is the difference between unity and the factor (1 -}- Cw{y, <po )). We can show later that the error involved in con- ; sidering the time phase as the space phase can easily reach 50 per cent or more, depending on the velocity spread of the electrons. 6. NUMERICAL CALCULATIONS Although the process of carrying out numerical computations has been discussed in Nordsieck's paper, it is desirable to recapitulate here I a few essential points including the new feature added. Using the work- ing equations (15), (16), (17) and (18), dai da 2 dw , dcp dy ' dy ' dy dy \ are calculable from ai , a^ , w and <p. The distance is divided into equal I intervals of A?/, and the forward integrations of Oi , ao , w and (p are per- f formed by a central difference formula ax{y + A?/) = ax{y) -f dy y+y2&y ■Ay In addition. d^ai dy^ and d 02 df in (17) are computed from the second difference formula such that d''ai - At/ _ dtti da\ dy^ j/=j/ \_dy y+l/2i,y dy y-^/2^y_ We thus calculate the behavior along the tube by forward integration j made in steps of Ay, starting from y = 0. At ?/ = the initial condi- tions are determined from Pierce's linearized theory. Because of its complications in notation, this will be discussed in detail in Appendix I. j Numerical calculations were carried out using the 701-type I.B.M. Table I a; U QC k c 6 Ml MJ ycsAT.) <! 01! H •i i a. 1 1 0.1 2.5 0.05 0.455 m max. 0.795662 -0.748052 5.6 1.26 0.415 2 0.1 2.5 0.1 0.541 Ml max. 0.827175 -0.787624 5.2 1.24 0.482 3 0.1 2.5 0.1 1.145 0.941;ui max. 0.778535 -1.05370 5.6 1.31 0.820 4 0.1 2.5 0.1 1.851 0.66jui max. 0.550736 -1.37968 7.0 1.36 1.05 J 5 0.1 2.5 0.2 0.720 m max. 0.900312 -0.873606 4.8 1.02 0.726 6 0.2 1.25 0.1 0.875 jui max. 0.769795 -1.04078 5.9 1.22 0.570 7 0.2 1.25 0.1 1.422 0.951^1 max. 0.724527 -1.29469 6.0 1.30 0.803 8 0.2 1.25 0.1 2.072 0.666mi max. 0.512528 -1.60435 7.6 1.35 1.08 9 0.2 2.5 0.05 0.765 Ml max. 0.731493 -0.973376 6.2 1.30 0.412 10 0.2 2.5 0.1 0.875 Ml max. 0.769795 -1.04078 5.8 1.22 0.490 11 0.2 2.5 0.1 1.422 0.941mi max. 0.724527 -1.29469 6.0 1.26 0.720 12 0.2 2.5 0.1 2.072 0.666mi max. 0.512528 -1.60435 7.2 1.25 0.92 13 0.2 2.5 0.1 2.401 0.300mi max. 0.230930 -1.76243 12.4 1.24 1.36 j U 0.2 2.5 0.15 0.976 Ml max. 0.812900 -1.10656 5.4 1.11 0.572 15 0.2 2.5 0.15 1.549 0.941mi max. 0.765101 -1.37540 5.8 1.14 1.03 16 0.2 2.5 0.15 2.2311 0.666mi max. 0.541234 -1.70180 7.0 1.12 1.22 17 0.2 2.5 0.15 2.575 0.300mi max. 0.243864 -1.86844 10.8 1.04 1.34 18 0.4 2.5 0.05 1.25 Ml max. 0.653014 -1.36746 7.6 1.26 0.315 19 0.4 2.5 0.1 1.38 Ml max. 0.701470 -1.47477 6.6 1.11 0.674 20 0.4 2.5 0.1 1.874 0.941mi max. 0.660223 -1.71341 7.8 1.19 1.05 21 0.4 2.5 0.1 2.458 0.666mi max. 0.467038 -1.99840 8.6 1.09 1.25 l> 360 A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 361 equipment. The problem was programmed by Miss D. C. Legaus. The cases computed are listed in Table I in which m and m2 are respectively Pierce's .xi and iji , and A,(d — iny) and tj at saturation will be discussed later. All the cases were computed with A^ = 0.2 using a model based on 24 electron discs per electronic wavelength. To estimate the error involved in the numerical work, Case (10) has been repeated for 48 elec- trons and Cases (10) and (19) for Ay = 0.1. The results obtained by using different numbers of electrons are almost identical and those ob- tained by varying the inter\'al A// indicate a difference in A (y) less than 1 per cent for Case (10) and about 6 per cent for Case (19). As error generally increases with QC and C the cases listed in this paper are limited to QC = 0.4 and C = 0.15. For larger QC or C, a model of more electrons or a smaller interval of integration, or both should be used. 7. POWER OUTPUT AND EFFICIENCY Define A(ij) = HVa,(yy + aM' -0(y)=i^n-'^-^ + by ^^^^ aiiy) We have then F{z,t) = ^A{y) cos ^ -^t- e{y) Uo (22) The power carried by the forward wave is therefore 2CA'hVo (23) (f) = \Z/o/ average and the efficiency is Eff. = ?£^^ = 2CA' or ^ = 2CA' (24) In Table I, the values of A(y), 6{y) and y at the saturation level are listed for every case computed. We mean by the saturation level, the distance along the tube or the value of y at which the voltage of the forward wave or the forward traveling power reaches its first peak. The Eff./C at the saturation level is plotted in Fig. 1 versus QC, for k = 2.5, h for maximum small-signal gain and C = small, 0.05, 0.1, 0.15 and 2. It is also plotted versus h in Fig. 2 for QC = 0.2, k = 2.5 and C = small, 0.1 and0.15, and in Fig. 3 for QC = 0.2, C = 0.1 and k = 1.25 and 2.50. In Fig. 2 the dotted curves indicate the values of h at Avhich 1 362 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G 4.5 0.5 Fig. 1 — The saturation eff./C versus QC, for k = 2.5, h for maximum small- signal gain and C = small, 0.1, 0.15 and 0.2. ixx = Ml (max), 0.94 jui(max), 0.67 iui(max) and 0.3 /ii(niax), respectively. It is seen that Eff./C decreases as C increases particularly when h is large. It is almost constant between k = 1.25 and 2.50 and decreases slowly for large values of C when QC increases. The (Eff./C) at saturation is also plotted versus QC in Fig. 4(a) for small C, and in Fig. 4(b) for C = 0.1. It should be noted that for C = 0.1 the values of Eff./C fall inside a very narrow region say between 2.5 to 3.5, whereas for small C they vary widely. 8, VELOCITY SPREAD In a traveling-wave amplifier, when electrons are decelerated by the circuit field, they contribute power to the circuit, and when electrons are accelerated, they gain kinetic energy at the expense of the circuit power. It is therefore of interest to plot the actual velocities of the fastest and the slowest electrons at the saturation level and find how they vary with the parameters QC, C, b and k. This is done in Fig. 5. These veloci- ties are also plotted versus y for Case 10 in Fig. 6, in which, the A(y) curve is added for reference. 9. THE BACKWARD WAVE AND THE FUNDAMENTAL COMPONENT OF THE ELECTRON CHARGE DENSITY Our calculation of efficiency has been based on the power carried by the forward wave only. One may, however, ask about the actual power A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 363 6.0 5.5 5.0 4.5 4.0 3.5 EFFI. C 3.0 (SAT.) 2.5 2.0 1 .5 1.0 0.5 1 QC = 0.2 1 1 A- — K=2.5 \ SMALI " *^r^ \ 1 Sa y^ 1 1 \ _/^ I Ji A 1 /^ \ 1 \ / \ t \ / \ \ ^ / \ \ X f \ ( \ \ \ \ ^ ' \ \ C = 0.1 '\ \ \ \ , lyj \ \ \ JT"^ \ \ V C=0.15 \ \ \ \ \ \ >"1 = 1 AX) /"1=C K 1 1 ).94/Z.(MAX) .at^ / 1 \^ /t/i = 0.67//i(MAX) \ //, = 0.3//i(MAX) 0.5 1.0 1.5 b 2.0 2.5 3.0 Fig. 2 — The saturation eff./C versus fe, for k = 2.5, QC = 0.2, and C = small, 0.1 and 0.15. The dotted curves indicate the values of h for m = \, 0.94, 0.67, and 0.3 of ;ui(max) respectively. output in the presence of the backward wave. For simphcity, we shall use the approximate solution (12b) which can be written in the form B{z, t) ^ Real Component of ZqIq c 4C 2(1 + hC) dax(y)Y ^ (da,{y)\- j^^-v,.-,y+j^\ (12d) with tan ^ = dij (laiiyT dy , dy dchiyY dy , As mentioned earlier that the complete solution of (6) is obtained by adding to (12b) a complementary function such that -yu 1+ r Qz ZqIq + c 4C 2(1 + bC) dy:) ^\dy ) ' -hy+ji (25) 364 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 EFFI. c (SAT.) 3 QC = o.2 C = 0.1 J<_=K25. 3- 2.50 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 b Fig. 3 — The saturation eff./C versus b, for QC = 0.2 C = 0.1 and k = 1.25 and 2.50. If the output circuit is matched by cold measurements, the backward wave must be zero at the output end, z = D. This determines Ci , that is, „ ZqIq c ^1 = ~^rPT or Cie jut+Toz 4C 2(1 + bC) ZqIq C //dai(t/)Y I / da2{y) Y ro(2+bc)D+ji dai{y)V / da2{y) y 4C 2(1 + 6C) y \ dy )z=o \ dy Jz^d (26) The backward wave therefore consists of two components. One compo- o 7 (a) C = SMALL ^- Ml = 0.67 /U,(MAX) D 5 ^^;;:^ ^^ EFFI. C 4 /U, = 0.94//i(MAX) ^^^ 1 (SAT.) ^ 2 ■"Zr^AtlC^AX) " (b) C = o.i = 0.94//, (MAX) 1 Xj fea,^_^-VZi = 0.67 /Z, (MAX) >U, = /i|(MAX)- 1 —===3 ^^^ 0.1 0.2 QC 0.3 0.4 0.1 0.2 QC 0.3 0.4 Fig. 4 — The saturation eff./C versus QC for h corresponding jui = 1, 0.94 and 0.67 of Mi(max), (a) for C = small, (b) for C = 0.1. A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 365 nent is coupled to the beam and has an amplitude equal to Zolo C IC 2(1 + bC) VX^'Y + K^y / \dy) which generally grows with the forward wave. It thus has a much larger amplitude at the output end than at the input end. The other component is a wave of constant amplitude, which travels in the direction opposite to the electron flow with a phase velocity equal to that of the cold cir- cuit. At the output end, 2 = Z), both components have the same ampli- tude but are opposite in sign. One thus realizes that there exists a re- flected wave of noticeable amplitude, in the form of (26), even though the output circuit is properly matched by cold measurements. Under j such circumstances, the voltage at the output end is the voltage of the forward wave and the power output is the power carried by the forward wave only. This is computed in (23). Since (26) is a cold circuit wave it may be eliminated by properly ad- c[-w], ■C[w], 5.0 4.5 4.0 3.5 5 3.0 2 o 9- 2.5 1.5 1.0 0.5 (a) ; / y / / ( r' ,.--- ( L"1 .-'■ (b) j^ V / / / y 1 Qw '"--^ ^-"^ (c) J i / / ^ / / ,''^ 1 / ( f r 1 1 1 1 < f 0.1 0.2 0.3 0.4 0.5 1.0 QC 1.5 2.0 2.5 b 0.05 0.10 0.15 0.20 Fig. 5 — Cw(y, <po) of the fast and the slowest electrons at the saturation level, (a) versus QC for k = 2.5, C = 0.1 and b for maximum small-signal gain; (b) versus 6 for A; = 2.50, C = 0.1 and QC = 0.2; and (c) versus C for A- = 2.50, QC = 0.2 and b for maximum small-signal gain. 366 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1950 3.5 3.0 2.5 2.0 9- ^1.5 U 1.0 0.5 > r\ / ^ \ / \ CASE 10 QC = 0.2 C = 0.1 b = 0.875 k = 2.5 MAXC(-W) / ,-' ''s \ \ /A(y) \ \ , 1 1 / /\ S // y / X- ./ ,^ "^AXCW i / / / y r ■7 / / A y ^-' ^ ^ :z=^ — ** — ^ 1.4 1.2 1.0 0.8 ID < 0.6 0.4 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 y Fig. 6 — Cw{y, (pa) of the fast and the slowest electrons versus y for Case (10). A{y) is also plotted in dotted lines for reference. justing the impedance of the output circuit. This may be necessary in practice for the purpose of avoiding possible regenerative oscillation. In doing so, the voltage at 2 = D is the sum of the voltage of the forward wave and that of the particular solution of the backward wave. In every case, the output power is always equal to the square of the net voltage actually at the output end divided by the impedance of the output cir- cuit. We find from (14), (15) and (16) that the fundamental component of electron charge density may be written as f s. \ h ( . dai{y) . da2(y)\ = Real component of 1/0 dai{y) dy , + doM dy (26) jo)—Toz—by+Ji ) where —Io/uq is the dc electron charge density, po . If (26) is compared with (12d) or (12c), it might seem surprising that the particular solution of the backward wave is just equal to the funda- A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 367 1.6 t.5 1.4 1.3 1.2 1.1 1.0 Pq 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.2 1.1 1.0 0.9 Pq 0.7 0.6 0.5 0.4 0.3 0.2 0.1 CASES 2, 10,19 (a) k = 2.5 - C = 0.1 b-»MAX n^ /' \ r \ J ' ^ r \ r\ \ / f V 1] s — QC=o.i/ / f 0.2 // r 0.4 // 1 V // \ 7 \ <^ L / \ A \ ^ ^ CASES 9, 0,14 (c) QC=0.2 - k=2.5 b-»MAX//| rv -\ 1 u \ r C=o,s/// \ / rf-o 10 \ ///o.05 i II k A f / ^ 8 4 y CASES 1C ,11,12 iH r V r\ (b) QC = o.2 C = o.i k = 2.5 r k\ (A \ // \ / y ^ c \ / \f A \ \ /^, = >U,MAx/^ ' / A / \ // / \ / f \ \ A / I / J /09« / 11 y / \ \ . // / / 11 / \J 17 // Ai.^i ^^ "w 1 /' y ^^ .^^ -^ >^ '^1 = 0.3X/,MAX 1 1 7 8 y 10 11 12 13 14 15 Fig. 7(a) — p^/po versus ?/, (a) using QC as the parameter, for A; = 2.5, C = 0.1, and 6 for maximum small-signal gain (Cases 2, 10, and 19) ; (b) using h as the param- eter, for k = 2.50, C = 0.1 and QC = 0.2 (Cases 10, 11, 12 and 13); and (c) using C as the parameter, for k = 2.50, QC = 0.2 and h for maximum small-signal gain (Cases 9, 10 and 14). 368 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 mental component of the electron charge density of the beam multiplied by a constant / Zq/o C 2uo 2wo\ h) (27) V 4C 2(1 + hC) The ratio of the electron charge density to the average charge density, P«(2) Po 2319^21 5 17/^,1 9 ^ +e Fig. 8(a) — y versus <f - hrj for QC = 0.2, k = 2.5, b for mi = 0.67 C = small. Ml (max) and A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 369 is plotted in Fig. 7 versus y, using QC, h and C, as the parameters. They lare also the curves for the backward wave (the component which is ! coupled to the beam) when multiplied by the proportional constant given in (27). It is interesting to see that the maximum values of p^/po are between 1.0 and 1.2 for QC = 0.2 and decrease as QC increases. The peaks of the curves do not occur at the saturation values of y. 10. y VERSUS ((p — by) diagrams To study the effect of C, b, and QC on efficiency y versus (<p — by) diagrams are plotted in Figs. 8(b), (c), (d) and (e) for Cases (21), (16), (10) and (21), respectively. {<p — by) here is ($ + 6) in Nordsieck's nota- tion. In these diagrams, the curves numbered from 1 to 24 correspond to the 24 electrons used in the calculation with each curve for one electron. Only odd numbered electrons are presented to avoid possible confusion arisen from too many lines. The reciprocal of the slope of the curve as -10 -9 -8 jo-by Fig. 8(b) — y versus <p C = 0.1 (Case 12). bij for QC = 0.2, k = 2.5, b for mi = 0.67Mi(max) and 370 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 given by (18) is proportional to the ac displacement of electron per unit of ij. (In small-C theorj^ it is proportional to the ac velocity of the elec- tron.) Concentration of curves is obviously proportional to the charge- density distribution of the beam. In the shaded regions, the axially di- rected electric field of the circuit is negative and thus accelerates elec- trons in the positive z direction. Electrons are decelerated in the un- shaded regions where the circuit field is positive. The boundaries of these regions are constant phase contours of the circuit wave. (They are con- stant $ contours in Nordsieck's notation.) These figures are actuallj' the "space-time" diagrams which unfold the historj^ of every electron from the input to the output ends. The effect of C can be clearly seen by comparing Figs. 8(a), (b) and (c). These diagrams are plotted for QC = 0.2, A; = 2.5, h for jui = 0.67 jui(max) and for Fig. 8(a), C = small, for Fig. 8(b), C = 0.1, and for Fig. 8(c), C = .15. It may be seen that because of the velocity spread of the electrons, the saturation level in Fig. 8(a) is 9.3 whereas in Figs. 8(b) and (c), it is 7.2 and 7.0, respectively. It is therefore not surprising that Eff./C decreases as C increases. The effects of h and QC may be observed by comparing Figs. 8(d) and (b), and Figs. 8(b) and (e), respectively. The details will not be de- scribed here. It is however suggested to study these diagrams with those given in the small-C theory. 7.2 5 1 23 9 11 i'5 7 3 1719 21 13 23 15 1719 21 ^" ^-^^ ?ny J^ V ^v:\ S \| A \- I SATURATION 6.8 6.4 6.0 5.6 6.2 .«,^ ^ *tf LEVEL " vK sL- ^ ^N ^ V \ ^ ■I ^ ^ L^ i ^ ^ y r \ rt 'a [ \ 1 rt ^VL / 1 V «x t / < $ ^W I / / -^ \ \ \ w t ll 1 '— T ^ kU\ 4.4 4.0 3.6 3.2 2.8 2.4 ?0 '^ t ^ \\\^ \ \ V \ I 1 / \\ 1 - 1 \ \ 1 \ 1 \ r- 1 /, 1 \ \\\ IS _H 1 1 li V-' \ \ \ ' % 1 ■ 1 1 1 1 1 i 1 > 1 1 \ \ 1 j i r 1 ll i i i5!r 1 ti23l 11 1 ,3 _ -1 9 in L. J 3 15 17 ig/sii 1 i i 23 -10 -9 -8 -7 -4 -3 1 10 Fig. 8(c) — y versus <p — by for QC = 0.2, k — 2.5, b for^i = 0.67^1 (max) and , C = 0.15 (Case 16). A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 371 11. A QUALITATIVE PICTURE AND CONCULSIONS We have exhibited in the previous sections the most important non- linear characteristics of the traveling wave ampUfier. Xumerical compu- tations based on a model of 24 electrons have been carried out for more than twenty cases covering useful ranges of design and operating parame- ters. The results obtained for the saturation Eff./C may be summarized as follows: (1) It decreases with C particularly at large values of QC. (2) For C = 0.1, it varies roughly from 3.7 for QC = 0.1 to 2.3 for i}C = 0.4, and only varies slightl}^ with h. (3) For C = 0.15, it varies from 2.7 to 2.5 for QC from 0.1 to 0.2 and \i corresponding to the maximum small-signal gain. It varies slightly with h for QC = 0.2. (4) It is almost constant between k — 1.25 and 2.50. In order to understand the traveling-wave tube better, it is important to have a simplified qualitative picture of its operation. It is obvious that to obtain higher amplification, more electrons must travel in the region where the circuit field is positive, that is, in the region where electrons 6.8 6.4 6.0 17 3 51 9i7 13 15 11 23 21 7 J 19 13 5 11 O^ N N cl. \ vV \ vn ^ . ^^vV ^ \ Vv \ \ ^A TURAT lOM ^ v^§^ \ \ \j l\ LEVEL 5.6 5.2 4.8 4.4 4.0 3.6 3.2 2.8 2.4 * 3" - . / ^ \ \ N / / / / 'V \\ \ K l\ \ 1 1 1 1 / / 1 /i^ \\ -^A "^ \ \ t / / 1 r 1 ■ /A\\\ f f fX V v 1 1 \ 1 I 1 / N \\v \\ \ / t / 1 \ ^ \° 1 I i / f \\\ f 1 / \ \ \ \ f 1 1 ( 1 r f // \\ 1 1 1 \\ / \ \ 11«13» 151 17 1 3 5 ] g\ inisl 15 7 19 21/ 2 3 ?n 1 1 1 1 \ li\ 1 i t J iJ -1 SP-by Fig. 8(d) — V versus <p — by for QC = 0.2, k = 2.5, b for m = mi (max) and '' = 0.1 (Case 10). 372 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 8.8 11 5 3 15 9 7 21 17? 23 1 15 21 23 ^-~ -^ R StiC^ 1 ^^ / /' 1- SATURATION 8.4 8.0 7.6 7.2 6.8 6.4 6.0 6.6 5.2 4.8 4.4 4.0 3.6 3.2 2.8 2.4 ?,0 IQ** :^ ^ ■" LEVEL ~ - 1 r4- ■~3 N, , / \ \ \ \, 7H 1 "^^ d \ / ^v.-— ^.-. ""^ \ \ t y- [V\ fe \ ^ \ >^ \ — p \ \ \ \ v\ I :: ''^: N\ "t ^ :\ \ \ -^^ K\l 1 H. '■.-■; 1 1 1 Ui y V t 1 J. 1 ^W ^\ / / )) r i t l-^i \V d \ r // 1 ( 1 i : 1 \\ ; 1 / i w \\ , :' ■,J; W I 1 1 1 — p- rl - 1 \ / ] \ -— - 1 1 1 1 \\\ 1 % 1 1 1 1 1 1 1 t n , 9 1 1 1 15; f ,'21 123 / !l V' 3 15 17 jl9 21 23 -1 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 : 3 3 - i 5 6 7 3 9 y-by Fig. 8(e) — y versus <p — by for QC = 0.4, k = 2.5, b for in = 0.67ui(max) and C = 0.1 (Case 21). are decelerated by the circuit field. At the input end of the tube, elec- trons are uniformly distributed both in the accelerating and decelerating field regions. Bunching takes place when the accelerated electrons push forward and the decelerated ones press backward. The center of a bunch of electrons is located well inside the decelerating field region because the circuit wave travels slower than the electrons on the average (6 is positive). The effectiveness of the amplification, or more specifically the ! saturation efficiency, therefore depends on (1), how tight the bunching :' is, and (2), how long a bunch travels inside the decelerating field region before its center crosses the boundary between the accelerating and decelerating fields. For small-C, the ac velocities of the electrons are small compared with the dc velocity. The electron bunch stays longer with the decelerating circuit field before reaching the saturation level when h or QC is larger. On the other hand, the space charge force, or large QC or k tends to dis- tort the bunching. As the consequence, the saturation efficiency increases , with h, and decreases as k or QC increases. When C becomes finite how- A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 373 ever, the ac velocities of the electrons are no longer small as compared I with their average speed. The velocity spread of the electrons becomes , an important factor in determining the efficiency. Its effect is to loosen the bunching, and consequently it lowers the saturation level and re- duces the limiting efficiency. It is seen from Figs. 5 and 6 that the . velocity spread increases sharply with C and also steadily with b and QC. \ This explains the fact that in the present calculation the saturation Eff./C decreases with C and is almost constant with h whereas in the 1 1 small-C theory it is constant with C and increases steadily with b. 12. ACKNOWLEDGEMENTS The writer wishes to thank J. R. Pierce for his guidance during the course of this research, and L. R. Walker for many interesting discus- sions concerning the working equations and the method of calculating I the backward wave. The writer is particularly grateful to Miss D. C. Leagus who, under the guidance of V. M. Wolontis, has carried out the ^ numerical work presented with endless effort and enthusiasm. APPENDIX The initial conditions at i/ = are computed from Pierce's linearized theory. For small-signal, we have ai(?/) = 4A(y) cos (6 -f ^2)2/ (A-1) «2(2/) = -4A(y) sin (6 + ju2)y (A-2) A(y) = ee"'' (A-3) Here e is taken equal to 0.03, a value which has been used in Tien-Walker- ' Wolontis' paper. Define ; ^ = wiy, <po) (A-4) 'X = pe-^'" + p*e^'^'> (A-5) dy where p* is the conjugate of p. After substituting (A-1) to (A-5) into the working equations (15) to (18) and carrying out considerable algebraic work, we obtain exactly Pierce's equation. 2 (1 + jC/i)(l + bC) innn \ ah \^ r\ r\ (j - >iCfi -h j}/ibC)(ti + jb) provided that + CO —k\((>(.y ,<po+<t>)—<p(.y ,Vo)l['^+Cw(.y ,ipo+(t>)] (A-7) • di^ sgn (^(?/, .i?o + «/)) - 9?(^, <Po)) = 8eQC (1 -f 3Cy){ii ^ jb) I e''" cos (arg [(1 -f jCm)(m + jb)] + my - ^0) 374 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Here ^ = Mi + JM2 or Pierce's rri + jiji . From (A-7) the value of Up is determined for a given QC. The ac velocities of the electrons are derived from (A-4), such as, = -26 M M + jb 1 + jcn e"^" cos ( arg M M + jb rvi+iCM/j + M22/ — <Po (A-8) (A-1), (A-2), (A-7) and (A-8) are the expressions used to calculate the initial conditions at y = 0, Avhen fn and jU2 are solved from Pierce's equa- tion (A-6). From (12c), the particular solution of the backward wave at small- signal is found to be j^, . ., -2iC(l+iC/x)(M+ib) ^Ml!/ 2j — CfjL -\- icb r [-2jC{\ - cos + iCM)(M+i6)' Cn + jcb + M2y — ^0 which agrees with Pierce's analysis 17 3. 4. REFERENCES 1. J. R. Pierce, Traveling-Wave Tubes, D. Van Nostrand Co., N.Y., 1950, p. 160. 2. R. L. Hess, Some Results in the Large-Signal Analysis of Traveling-Wave Tubes, Technical Report Series No. 60, Issue No. 131, Electronic Research Laboratory, University of California, Berkeley, California. C. K. Birdsall, unpublished work. J. J. Caldwell, unpublished work. 5. P. Parzen, Nonlinear Effects in Traveling-Wave Amplifiers, TR/AF-4, Radia- tion Laboratory, The Johns Hopkins University, April 27, 1954. 6. A. Kiel and P. Parzen, Non-linear Wave Propagation in Traveling-Wave Amplifiers, TR/AF-13, Radiation Laboratory, The Johns Hopkins Univer- sity, March, 1955. 7. A. Nordsieck, Theory of the Large-Signal Behavior of Traveling-Wave Ampli- fiers, Proc. I.R.E., 41, pp. 630-637, May, 1953. H. C. Poulter, Large Signal Theory of the Traveling-Wave Tube, Tech. Re- port No. 73, Electronics Research Laboratory, Stanford University, Cali- fornia, Jan., 1954. P. K. Tien, L. R. Walker and V. M. Wolontis, A Large Signal Theory of Trav- eling-Wave Amplifiers, Proc. LR.E., 43, pp. 260-277 March, 1955. J. E. Rowe, A Large Signal Analysis of the Traveling-Wave Amplifier, Tech. Report No. 19, Electron Tube Laboratory, University of Michigan, Ann Arbor, April, 1955. 11. P. K. Tien and L. R. Walker, Correspondence Section, Proc. I.R.E., 43, p. 1007, Aug., 1955. Nordsieck, op. cit., equation (1). L. Brillouin, The Traveling-Wave Tube (Discussion of Waves for Large Amplitudes), J. Appl. Phys., 20, p. 1197, Dec, 1949. Pierce, op. cit., p. 9. Nordsieck, op. cit., equation (4). Pierce, op. cit., equation (7.13). 17. J. R. Pierce, Theory of Traveling-Wave Tube, Appendix A, Proc. I.R.E. 35, p. 121, Feb., 1947. 8. 10 12 13 14 15 16 A Detailed Analysis of Beam Formation with Electron Guns of the Pierce Type By W. E. DANIELSON, J. L. ROSENFELD,* and J. A. SALOOM (Manuscript received November 10, 1955) The theory of Cutler and Hines is extended in this paper to permit an analysis of heam-spreading in electron guns of high convergence. A lens correction for the finite size of the anode aperture is also included. The Cutler and Hines theory was not applicable to cases where the effects of thermal velocities are large compared with those of space charge and it did not include a lens correction. Gun design charts are presented which include all of these effects. These charts may he conveniently used in choosing design parameters to produce a prescribed beam. CONTENTS 1 . Introduction 377 2. Present Status of Gun Design; Limitations 378 3. Treatment of the Anode Lens Problem 379 A. Superposition Approach 379 B. Use of a False Cathode 382 C. Calculation of Anode Lens Strength by the Two Methods 383 4. Treatment of Beam Spreading, Including the Effect of Thermal Electrons 388 A. The Gun Region 388 B. The Drift Region 392 5. Numerical Data for Electron Gun and Beam Design 402 A. Choice of Variables 402 B. Tabular Data 402 C. Graphical Data, Including Design Charts and Beam Profiles 402 D. Examples of Gun Design Using Design Charts 403 6. Comparison of Theory with Experiment 413 A. Measurement of Current Densities in the Beam 413 B. Comparison of the Experimentally Measured Spreading of a Beam with that Predicted Theoretically 416 C. Comparison of Experimental and Theoretical Current Density Distri- butions where the Minimum Beam Diameter is Reached 418 D. Variation of Beam Profile with T 418 7. Some Additional Remarks on Gun Design 418 * Mr. Rosenfeld participated in this work while on assignment to the Labora- tories as part of the M.I.T. Cooperative Program. 375 376 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 GLOSSARY OF SYMBOLS Ai , 2 anode designations B, C anode potentials Ci , 2 functions used in evaluating cr+' dA increment of area dl, dz increments of length e . electronic charge, base of natural logarithms En electric field normal to electron path F modified focal length of the anode lens Fd focal length of the anode lens as given by Davisson^ Fn force acting normal to an electronic path Fr , a fraction of the total current which would flow through a circle of radius r, a /, Id total beam current It beam current within a radius, r, of the center J current density k Boltzman's constant K - a quantity proportional to gun perveance m electronic mass P gun perveance P{r) probability that a thermal electron has a radial posi- tion between r and r -\- dr r radial distance from beam axis Va , c anode, cathode radii r^ distance from beam axis to path of an electron emitted with zero velocity at the edge of the cathode rgs radius of circle through which 95% of the beam cur- rent would pass f distance from center of curvature of cathode; hence, fc is the cathode radius of curvature and (fc — fa) is the distance from cathode to anode re+' slope of edge nonthermal electron path on drift side of enode lens Te-' slope of edge nonthermal electron path on gun side of anode lens R a dummy integration variable t time T cathode temperature in degrees K u longitudinal electron velocity Vc , X , y transverse electron velocities V, Va , f , X beam voltages with cathode taken as ground BEAM FORMATION WITH ELECTRON GUNS 377 V(f, /■), Vc.(f, potential distributions used in the anode lens study r), etc. V' voltage gradient z distance along the beam from the anode lens 2n,in distance to the point where rgs is a minimum ( — a) Langmuir potential parameter for spherical cathode- anode gun geometry 7 slope of an electron's path after coming into a space charge free region just beyond the anode lens r the factor which divides Fd to give the modified anode focal length 5 dimensionless radius parameter €o dielectric constant of free space f dimensionless voltage parameter 6 slope of an electron's path in the gun region r} charge to mass ratio for the electron fx normalized radial position in a beam a the radial position of an electron which left the cathode center with "normal" transverse velocity (T+' slope of o--electron on drift side of anode lens a J slope of (T-electron on gun side of anode lens ^ electric flux 1. INTRODUCTION During the past few years there have been several additions to the family of microwave tubes rec}uiring long electron beams of small diame- ter and high current density. Due to the limited electron current which can be "drawn from unit area of a cathode surface with some assurance of long cathode operating life, high density electron beams have been produced largely through the use of convergent electron guns which increase markedly the current density in the beam over that at the cathode surface. An elegant approach to the design of convergent electron guns was provided by J. R. Pierce^ in 1940. Electron guns designed by this method are known as Pierce guns and have found extensive use in the produc- tion of long, high density beams for microwave tubes. ]\Iore recent studies, reviewed in Section 2, have led to a better under- standing of the influence on the electron beam of (a) the finite velocities with which electrons are emitted from the cathode surface, and (b) the defocusing electric fields associated with the transition from the ac- celerating region of the gun to the drift region beyond. Although these two effects have heretofore been treated separately, it is in many cases 378 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 necessary to produce electron beams under circumstances where both effects are important and so must be dealt with simultaneously and more precisely than has until now been possible. It is the purpose of this paper to provide a simple design procedure for typical Pierce guns which in- cludes both effects. Satisfactory agreement has been obtained between measured l^eam contours and those predicted for several guns having per\'eances (i.e., ratios of beam current to the ^^ power of the anode voltage) from 0.07 X 10-« to 0.7 X 10"^ amp (volt)-3/2. 2. PRESENT STATUS OF GUN DESIGN — LIMITATIONS Gun design techniques of the type originally suggested by J. R. Pierce were enlarged in papers by SamueP and by Field^ in 1945 and 1946. Samuel's work did not consider the effect of thermal velocities on beam shape and, although Field pointed out the importance of thermal veloci- ties in limiting the theoretically attainable current density, no method for predicting beam size and shape by including thermal effects was suggested. The problem of the divergent effect of the anode lens was treated in terms of the Davisson"* electrostatic lens formula, and no corrections were applied.* More recently. Cutler and Hines^ and also Cutler and Saloom^ have presented theoretical and experimental work which shows the pro- nounced effects of the thermal velocity distribution on the size and shape of beams produced by Pierce guns. Cutler and Saloom also point to the critical role of the beam-forming electrode in minimizing beam distor- tion due to improper fields in the region where the cathode and the beam-forming electrode would ideally meet. With regard to the anode lens effect, these authors also show experimental data which strongly suggest a more divergent lens than given by the Davisson formula. The Hines and Cutler thermal velocity calculations have been used"' "^ to predict departures in current density from that which should prevail in ideal beams where thermal electrons are absent. Their theory is limited, however, by the assumption that the beam-spreading caused by thermal velocities is small compared to the nominal beam size. In reviewing the various successes of the above mentioned papers in affording valuable tools for electron beam design, it appeared to the present authors that significant improvement could be made, in two respects, by extensions of existing theories. First, a more thorough in- * It is in fact erroneously statoci in Reference 5 that the lens action of an actual structure must be somewhat weaker than i)re(licted by the Davisson formula so that the beam on leaving the anode hole is more convergent than would be calcu- lated by llie Davisson method. This cjuestion is discussed further in Section 3. BEAM FORMATION WITH ELECTRON GUNS 379 vestigation of the anode lens effect was called for; and second, there was a need to extend thermal velocity calculations to include cases where the percentage increase in beam size due to thermal electrons was as large as 100 per cent or 200 per cent. Some suggestions toward meeting this second need have been included in a paper by M. E. Hines.* They have been applied to two-dimensional beams by R. L. Schrag.^ The particular assumptions and methods of the present paper as applied to the two needs cited above are somewhat different from those of Refer- ences 8 and 9, and are fully treated in the sections which follow. 3. TREATMENT OF THE ANODE LENS PROBLEM Using thermal velocity calculations of the type made in Reference 6, it can easily be shown that at the anode plane of a typical moderate perveance Pierce type electron gun, the average spread in radial posi- tion of those electrons which originate from the same point of the cathode is several times smaller than the beam diameter. For guns of this type, then, we may look for the effect of the anode aperture on an electron beam for the idealized case in which thermal velocities are absent and confidently apply the correction to the anode lens formula so obtained to the case of a real beam. Several authors have been concerned with the diverging effect of a hole in an accelerating electrode where the field drops to zero in the space beyond, ^° but these treatments do not include space charge effects except as given by the Davisson formula for the focal length, Fd , of the lens: F. = -^ (1) where V would be the magnitude of the electric field at the aperture if it were gridded, and V would be the voltage there. In attempting to describe the effect of the anode hole with more ac- curacy than (1) affords, we have combined analytical methods with electrolytic tank measurements in two i-ather different ways. The first method to be given is more rigorous than the second, hut a modification of the second method is much easier to use and gives essentially the same result. A. Siipcrposition Approach to the Anode Lens Problem Special techniques are required for finding electron trajectories in a space charge limited Pierce gun having a non-gridded anode. M. E. 380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Hines has suggested* that a fairly accurate description of the potential distribution in such guns can be obtained by a superposition method as follows: By the usual tank methods, find suitable beam forming electrode and anode shapes for conical space charge limited flow in a diode having! cathode and anode radii of curvature given by fc and f„i , respectively, as shown in Fig. 1(a). Using the electrolytic tank with an insulator along the line which represents the beam edge, trace out an equipotential which intersects the insulator at a distance fa2 from the cathode center of curvature. Let the cathode be at ground potential and let the voltage on anode Ai be called B. Suppose, now, that we are interested in electron trajectories in a non-gridded gun where the edge of the anode hole is a distance fai from the center of curvature of the cathode. Let the voltage, C, for this anode be chosen the same as the value of the equipotential traced out above for the case of cathode at ground potential and A\ at potential B. If we consider the space charge limited flow from a cathode which is followed by the apertured anode, Ai , and the full anode, Ai , at potentials C and B, respectively, it is clear that a conical flow of the type which would exist between concentric spheres will re- sult. The flow for such cases was treated by Langmuir,^ and the associ- ated potentials are commonly called the "Langmuir potentials." If we operate both Ai and A2 at potential C, however, the electrons will pass through the aperture in anode A2 into a nearly field-free region. . If the distance, fa2 — Tai , from A2 to Ai is greater than the diameter of the aperture in A2 , the flow will depend very little on the shape of Ai and the electron trajectories and associated equipotentials will be of the type we wish to consider except in a small region near Ai . We will shortly make use of the fact that the space charge between cathode and A 2 is not changed much when the voltage on Ai is changed from B to C, but first we will define a set of potential functions which will be needed. In order to obtain the potential at arbitrary points in any axially sym- metric gun when space charge is not neglected, w^e may superpose po- tential solutions to 3 separate problems where, in each case, the boundary condition that each electrode be an equipotential is satisfied. We will follow the usual notation in using f for the distance of a general point from the cathode center of curvature, and r for its radial distance from the axis of symmetry. Let Vdr, r), Vh(r, >') and Vsdr, r) be the three potential solutions where: (1) Vaif, r) is the solution for the case of no space charge with Ai and cathode at zero potential and A 2 at potential C, (2) Vb{r^ r) is the solution for the case of no space charge with A2 * Verbal disclosure. BEAM FORMATION WITH ELECTRON GUNS 381 and cathode at zero potential and Ai at potential B, and (3) Vsc(f, r) is the soUition when space charge is present but when Ax , A^ , and cathode are all grounded. If the configuration of charge which contributes to Vs<-(f, r) is that corresponding to ideal Pierce type flow, then we can use the principle of superposition to give the Langmuir potential, VL(r, r): VUr, r) = Vcif, r) + V,{f, r) + V..{f, r) (2) Furthermore, the potential configuration for the case where ^i and A2 are at potentical C can be written V =V.-\-^V, + F(.c)' (3) where the functional notation has been dropped and F(sc)' is the po icntial due to the new space charge when Ai and A2 are grounded. We are now ready to use the fact that F(sc)' may be well approximated 1)3' Fsc which is easily obtained from (2). This substitution may be justified by noting that the space charge distribution in a gun using a \'oltage C for Ai does not differ significanth^ from the corresponding dis- tribution when Ai is at voltage B except in the region near and beyond A-i where the charge density is small anyway (because of the high electron velocities there). Substituting Fsc as given by (2) for F(sc)' in (3) then gives V Vi 1 B, V, (4) We have thus obtained an expression, (4), for the potential at an arbi- ANODE A2 v=c ANODE A, V = B CATHODE Fig. 1(a) — ■ Electrode configuration for anode lens evaluation in Section 2>A. 382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ■i trary point in our gun in terms of the well known solution for space charge limited flow between two concentric spheres, Vl , and a potential distribution, Vb , which does not depend on space charge and can there- fore be obtained in the electrolytic tank. Once the potential distribution is found, electron trajectories may be calculated, and an equivalent lens sj^stem found. Equation (4) is used in this way in Part C as one basis for estimating a correction to the Davisson equation. (It will be noted that i (4) predicts a small but finite negative field at the cathode. This is be- cause the space charge density associated with Fsc is slightly greater near the cathode than that associated with F(sc)' , and it is this latter space charge which will make the field zero at the cathode under real space charge limited operation. Equation (4), as applied in Part C of this section, is used to give the voltage as a function of position at all points except near the cathode where the voltage curves are extended smoothly to make the field at the cathode vanish.) B. Use of a False Cathode in Treating the Anode Lens Problem Before evaluating the lens effect by use of (4), it will be useful to de- velop another approach which is a little simpler. The evaluation of the lens effect predicted by both methods will then be pursued in Part C where the separate results are compared. In Part A we noted that no serious error is made in neglecting the dif- ference between the two space charge configurations considered there because these differences were mainly in the very low space charge region near and beyond A2 . It similarly follows that we can, with only 1 a small decrease in accuracy, ignore the space charge in the region near and beyond A2 so long as we properly account for the effect of the high space charge regions closer to the cathode. To place the foregoing obser- vations on a more quantitative basis, we may graph the Langmuir po- tential (for space charge limited flow between concentric spheres) versus the distance from cathode toward anode, and then superpose a plot of the potential from LaPlace's equation (concentric spheres; no space charge) which will have the same value and slope at the anode. The La- Place curve will depart significantly from the Langmuir in the region of the cathode, but will adequately represent it farther out." Our experi- ence has shown that the representation is "adequate" until the difference between the two potentials exceeds about 2 per cent of the anode voltage. Then, since space charge is not important in the region near the anode for the case of a gridded Pierce gun, corresponding to space charge limited flow between concentric spheres, it can be expected to be similarly unimportant for cases where the grid is replaced by an aperture. Let us I BEAM FORMATION WITH ELECTRON GUNS 383 therefore consider a case where electrons are emitted perpendicularly and with finite velocity from what would be an appropriate spherical equipotential between cathode and anode in a Pierce type gun. So long as (a) there is good agreement between the LaPlace and Langmuir curves at this artificial cathode and (b) the distance from this artificial cathode to the anode hole is somewhat greater than the hole diameter, we will liiid that the divergent effect of the anode hole will be very nearly the same in this concocted space charge free case as in the actual case where space charge is present. (The quantitative support for this last state- ment comes largely from the agreement between calculations based on this method and calculations by method A.) The electrode configura- tion is shown in Fig. 1(b), and the potential distribution in this space charge free anode region can now be easily obtained in the electrolytic j tank. This potential distribution will be used in the next section to pro- ^•ide a second basis for estimating a correction to the Davisson equation. C. Calculation of Anode Lens Strength by the Two Methods The Davisson equation, (1), may be derived by assuming that none of the electric field lines which originate on charges in the cathode-anode region leave the beam before reaching the ideal anode plane where the voltage is F, and that all of these field lines leave the beam symmetrically and radially in the immediate neighborhood of the anode. Electrons I are thus considered to travel in a straight line from cathode to anode, and then to receive a sudden radial impulse as they cross radially diverg- ing electric field lines at the anode plane. A discontinuous change in CATHODE ANODE A2 V = C ANODE A, v = c (b) ^ FALSE CATHODE Fig. 1(b) — The introduction of a false cathode at the appropriate potential lUows the effect of space charge on the potential near the anode hole to be satis- :ictorily approximated as discussed in Section 3i?. 384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 slope is therefore produced as is common to all thin lens approximations. The diverging effect of electric field lines which originate on charges which have passed the anode plane is then normally accounted for by the universal beam spread curve/" In our attempt to evaluate the lens effect more accurately, we will still depend upon using the universal beam spread curve in the region following the lens and on treating the ; equivalent anode lens as thin. Consequently our improved accuracy must come from a mathematical treatment which allows the electric field lines originating in the cathode-anode region to leave the beam grad- ually, rather than a treatment where all of these flux lines leave the beam , at the anode plane. In practice the measured perveances, P(= I/V^'^), of active guns of the type considered here have averaged within 1 or 2 per cent of those predicted for corresponding gridded Pierce guns. There- fore the total space charge between cathode and anode is much the same with and without the use of a grid, even though the charge dis- tribution is not the same in the two cases. The total flux which must leave our beam is therefore the same as that which will leave the cor- , responding idealized beam and we may write yp = I EndA = TT/VFidea/ (5) w^here En is the electric field normal to the edge of the beam, ra = rdfa/fc) is the beam radius at the anode lens, and Videai is the magnitude of the field at the corresponding gridded Pierce gun anode. To find the appropriate thin lens focal length we will now find the total integrated transverse impulse which would be given to an elec- tron which follows a straight-line path on both sides of the lens (see Fig. 2), and we will equate this impulse to wAw where An is the transverse velocity given to the electron as it passes through the equivalent thin lens. In this connection we will restrict our attention to paraxial elec- trons and evaluate the transverse electric fields from (4) and from the tank plot outlined in Section B, respectively. The total transverse im- pulse experienced by an electron can be written f Fn dt = e [ —dl (()) J Path J Path U where u is the velocity along the path and Fn is the force normal to the path. We will usually find that the correction to (1) is less than about 20 per cent. It will therefore be worthwhile to put (6) in a form which in effect allows us to calculate deviaiions from Fu as given by (1) instead BEAM FORMATION WITH ELECTRON GUNS 385 1 of deriving a completely new expression for F. In accomplishing this piir- f pose, it will be helpful to define a dimensionless function of radius, 6, by - = 1 + 5, r and a dimensionless function of voltage, f, by (7a) (7b) where Ta is the radius at the anode lens when the lens is considered thin, and T^'x is a constant voltage to be specified later. (Note that the quan- tities 5 and f are not necessarily small compared to 1.) Using u = \/2r]V, and substituting for -y/V from (7b) we obtain f En dl 4 r , = 7~7tW / ^"^1 + r + 5 + rs) ^z (8) where use has also been made of (7a) in the form 1 = r(l + d)/ra . Now, as outlined above, we equate this impulse to 771 An, and we obtain ^» = WW. (/ ''■'' '' + / ''"'■'^ + ' + ^'' 'i (9) CATHODE Fig. 2 — The heavy line represents an electron's path when the effect of the .•mode hole may be represented by a thin lens, and when space charge forces are iihsent in the region following the anode aperture. For paraxial electrons, the (negative) focal length is related to the indicated angles by (y = + Ta/F). 386 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CENTER OF ~~ CURVATURE OF CATHODE SURFACE Fig. 3 — The gun parameters used in Section SC for comparing two methods of evaluating the effect of the anode lens. The first integral can be obtained from (5) ; hence, if we are able to choose Vx so that the second integral vanishes, we may write: Au = raV'2riVx The reciprocal of the thin lens focal length is therefore i _ ^ _ ^' F ~ ~raUf ^ ~^VWf (10) where w/ and F/ are the final velocity and voltage of the electron after it leaves the lens region. The real task, then, is to use the potential distribution in the gun as obtained by the methods of Part A or Part B above to find the value of V X which causes the last integral in (9) to vanish : To compare the two focal lengths obtained by the methods of Part A and B respectively, a specific tank design of the type indicated in Fig. 1 was carried out. The relevant gun parameters are indicated in Fig. 3. Approximate voltages on and near the beam axis were obtained as indicated in Parts A and B, above, with the exception that in the superposition method, A, special techniques were used to subtract the effect of the space charge lying in the post-anode region (because the effect of this space charge is accounted for separately as a divergent force in the drift region*). From these data, * See Section 4B. BEAM FOKMATION WITH ELECTRON GUNS 387 800 805 810 815 820 825 830 835 840 845 850 855 860 Fig. 4 — Curves for finding the value of Fx to be used in equation (10) for the set of gun parameters of Fig. 3. l)oth the direction and magnitude of the total electric field near the beam axis were (with much labor) determined. Once these data had been obtained, a trial value was selected for Vx , and the corresponding local length was calculated by (10). This enabled the electron's path through the associated thin lens to be specified so that, at this point in the procedure, both r and V were known functions of ^, and the quan- tities 8 and f were then obtained as functions of € from (7). Finally the second integral in (9) was evaluated for the particular Vx chosen, and then the process was repeated for other values of Vx . Fig. 4 shows curves whose ordinates are proportional to this second integral and whose abscissae are trial values for Vx . As noted above, the appropriate value for Vx is that value which makes the ordinate vanish, so that we obtain T'c = 813 and 839 for methods A and B, respectively. The percentage difference in the focal lengths obtained by the two methods is thus only 1 .6 per cent, and the reasonableness of making calculations as outlined in Part B is thus put on a more quantitative basis. Even calculations based on the method of Part B are tedious, and we naturally look for simpler methods of estimating the lens effect. In this fonnection we have found that Vx is usually well approximated by the \alue of the potential at the point of intersection between the beam axis and the ideal anode sphere. The specific values of the potential at this point as obtained by the methods of Parts A and B were 814 and 827, respectively. It will be noted that these values agree remarkably well with the values obtained above. Furthermore, very little extra effort is required to obtain the potential at this intersection in the false cathode case: I 388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Electrolytic tank measurements are normally made in the cathode- anode region to give the potential variation along the outside edge of the electron beam (for comparison with the Langmuir potential) ; hence, by tracing out a suitable equipotential line, the shape of the false cathode can easily be obtained. With the false cathode in place and at the proper potential, the approximate value for Vx is then obtained by a direct tank measurement of the potential at an axial point whose distance from the true cathode center is (fc — fa) as outlined above. Although finite elec- tron emission velocities typically do not much influence the trajectory of an electron at the anode, they do nevertheless significantly alter the beam in the region beyond. It is in this affected region where experi- mental data can be conveniently taken. We must, therefore, postpone a comparison of lens theory with experiment until the effect of thermal velocities has been treated. At that time theoretical predictions com- bining the effects of both thermal velocities and the anode lens can be made and compared with experiment. Such a comparison is made in Section 6. 4. TREATMENT OF BEAM SPREADING, INCLUDING THE EFFECT OF THERMAL ELECTRONS Jn Section 2 the desirability of having an approach to the thermal spreading of a beam which would be applicable under a wide variety of conditions was stressed. In particular, there was a need to extend ther- mal velocity calculations to include the effects of thermal velocities even when electrons with high average transverse velocities perturb the beam size by as much as 100 or 200 per cent. Furthermore, a realistic mathe- matical description which would allow electrons to cross the axis seemed essential. The method described below is intended adequately to answer these requirements. A. The Gun Region The Hines-Cutler method of including the effect of thermal velocities on beam size and shape leads one to conclude that, for usual anode voltages and gun perveance, the beam density profile in the plane of the anode hole is not appreciably altered by thermal velocities of emis- sion. (This statement will be verified and put on a more quantitative basis below.) Under these conditions, the beam at the anode is ade- quately described by the Hines-Cutler treatment. We will therefore find it convenient to adopt their notation where possible, and it will be worthwhile to review their approach to the thermal problem. BEAM FORMATION WITH ELECTRON GUNS 389 It is assumed that electrons are emitted from the cathode of a therm- ionic gun with a IMaxwelhan distribution of transverse velocities ZTTfC 1 where Jc is the cathode current density in the z direction, T is the cath- jode temperature, and v^: and Vy are transverse velocities. The number iof electrons emitted per second with radially directed voltages between V and V + dV is then -(.Ve/kT) (S) ^J. = /.e— -^^^(^^j (12) Now in the accelerating region of an ideal Pierce gun (and more generally I in any beam exhibiting laminar flow and having constant current density ()\'er its cross section) the electric field component perpendicular to the axis of symmetry must vary linearly with radius. Conseciuently Hines and Cutler measure radial position in the electron beam as a fraction, ^, of the outer beam radius (re) at the same longitudinal position, r = fire (13) The laminar flow assumption for constant current densities and small beam angles implies a radius of curvature for laminar electrons which so varies linearly with radius at any given cross section so that a Substituting for r from (13), (14) becomes rfV , /2 dre\ dfj. d^^VcTt)dt=^ ^^^^ where Ve and dr /dt can be easily obtained from the ideal Langmuir solution. Since the eciuation is linear in /x, we are assured that the radial position of a non-ideal electron that is emitted with finite transverse velocity from the cathode center (where ^ = 0) will, at any axial point, be proportional to dii/dt at the cathode. Let us now define a quantity "o-" such that n = a/re is the solution to (15) with the boundary conditions /Xr = and _ 1 where the subscript c denotes evaluation at the cathode surface, k is 390 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Boltzman's constant, T is the cathode temperature in degrees Kelvin, and m is mass of the electron. For the case ixc = 0, but with arbitrary initial transverse velocity, we will then have /^\ ^^nl_ /kf ^^^'^ Tc y m Plence we can express a in terms of the thermal electron's radial po- sition (r), and its initial transverse velocity, Vc , y m _ y . . - . /kT dt } f The quantity a can now be related to the radial spread of thermal electrons (emitted from a given point on the cathode) with respect to an electron with no initial velocity: By (11) we see that the number of electrons leaving the cathode with dji/dt = Vc/ve is proportional to Vc exp —Vcm/2kT. Suppose many experiments were conducted where all electrons except one at the cathode center had zero emission velocity, and suppose the number of times the initial transverse velocity of the single thermal electron were chosen as Vc , is proportional to Vc exp — Vcm/2kT. Then the probability, P{r), that the thermal electron would have a radial position between r and r -\- dr when it arrived at the transverse plane of interest would be proportional to Vc exp —Vc^(m/2kT). Here Vc is the proper transverse velocity to cause arrival at radius r, and by (17) we have a y m so that the probability becomes Pir) = J.e-^^'''-'^ d (^Q (18) We therefore identify cr with the standard deviation in a normal or Gaussian distribution of points in two dimensions. At the real cathod(\ thermal electrons are simultaneously being emitted from the cathode surface with a range of transverse velocities. However, if a as definml above is small in comparison with r,. , the forces experienced by a ther- mal electron when other thermal electrons are present will be very nearly BEAM FORMATION WITH ELECTRON GUNS 391 2.0 1.8 1.6 > 1.4 t 1.2 \%y 1.0 0.8 0.6 0.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Fig. 5 — Curves useful in finding the transverse displacement of electron tra- i jectories at the anode of Pierce-type guns. i tlie same as the forces involved in the equations above. Thus if o- <3C J'e , (18) may be taken as the distribution, in a transverse plane, of those electrons which were simultaneously emitted at the cathode center. I Furthermore, the nature of the Pierce gun region is such that electrons emitted from any other point on the cathode will be similarly distributed \\ ith respect to the path of an electron emitted from this other point w ith zero transverse velocity (so long as they stay within the confines , of the ideal beam). Hines and Cutler have integrated (15) with n^ = ' and {dn/dt)c = 1 to give g/ {fc\/kT/'2eV^ at the anode as a function of ; /", /fo . This relationship is included here in graphical form as Fig. 5. , For a large class of magnetically shielded Pierce-type electron guns, including all that are now used in our traveling wave tubes, Ve/a at the anode is indeed found to be greater than 5 (in most cases, greater than 10) so that evaluation of a at the anode of such guns can be made with considerable accuracy by the methods outlined above. One source of error lies in the assumption that electrons which are emitted from a point at the cathode edge become normally distributed about the cor- responding non-thermal (no transverse velocity of emission) electron's path, and with the same standard deviation as calculated for electrons from the cathode center. In the gun region where Ve/a tends to be large this difference between representative a- values for the peripheral and central parts of the beam is unimportant, but it must be re-examined in tlie drift region following the anode. 392 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 We have already investigated the region of the anode hole in some detail in Section 3 and have found it worth while to modify the ideal Davisson expression for focal length of an equivalent anode lens. In particular, let us define a quantity F by F = focal length = Fd/T (19) where Fd is the Davisson focal length. Thus T represents a corrective factor to be applied to Fd to give a more accurate value for the focal length. In so far as any thin lens is capable of describing the effects of diverging fields in the anode region, we may then use the appropriate optical formulas to transfer our knowledge of the electron trajectories (calculated in the anode region as outlined above) to the start of the drift region. In particular, -f (20) where {dr/dz)i and {dr/dz)^ are the slopes of the path just before and just after the lens, and r is the distance from the axis to the point where the ideal path crosses the lens plane. B. The Drift Region Although Te/a- was found to be large at the anode plane for most guns of interest, this ratio often shrinks to 1 or less at an axial distance of only a few beam diameters from the lens. Therefore, the assumption that electron trajectories may be found by using the space charge forces which would exist in the absence of thermal velocities of emission (i.e., forces consistant with the universal beam spread curve) may lead to very appreciable error. For example, if ecjual normal (Gaussian) distributions of points about a central point are superposed so that the central points are equally dense throughout a circle of radius Te , and if the standard de- viation for each of the normal distributions is cr = r^ , the relative density of points in the center of the circle is only about 39 per cent of what it would be Avith a < (re/5). In order to minimize errors of this type we have modified the Hines- Cutler treatment of the drift space in two ways: (1) The forces influenc- ing the trajectories of the non- thermal electrons are calculated from a progressive estimation of the actual space charge configuration as modi- fied by the presence of thermal electrons. (2) Some account is taken of the fact that, as the space charge density in the beam becomes less uni- form as a function of radius, the spread of electrons near the center of the beam increases more rapidly than does the corresponding spread BEAM FORMATION WITH ELECTRON GUNS 393 farther out. Since item (1) is influenced by item (2), the specific as- sumptions involved in the latter case will be treated first. When current density is uniform across the beam and its cross section changes slowly with distance, considerations of the type outlined above for the gun region show that those thermal electrons which remain within the beam will continue to have a Gaussian distribution with re- spect to a non-thermal electron emitted from the same cathode point. When current density is not uniform over the cross section, we would still like to preserve the mathematical simplicity of obtaining the current density as a function of beam radius merely by superposing Gaussian distributions which can be associated with each non-thermal electron. To lessen the error involved in this simplified approach, we will arrive at a value for the standard deviation, a (which specifies the Gaussian distribution), in a rather special way. In particular, a at any axial po- sition, z, will be taken as the radial coordinate of an electron emitted from the center of the cathode with a transverse velocity of emission given by, ve = y- — (21) m It is clear from (17) that for such an electron, r = o- in the gun region. From (18), the fraction of the electrons from a common point on the cathode which will have r ^ a in the gun region is 2 fraction = [ e'^'-'"-''^ d ^= I - e'"' = 0.393 (22) If re denotes the radial position of the outermost non-thermal electron and if 0- > /■,, , the "a--electron" will be moving in a region where the space charge density is significantly lower than at the axis. We could, of course, have followed the path of an electron with initial velocity equal to say 0.1 or 10 times that given in (21) and called the correspond- nig radius O.lcr or lOo-. The reason for preferring (21) is that about 0.4 or nearly half of the thermal electrons emitted from a common cathode point will have wandered a distance less than a from the path of a non- thermal electron emitted from the same cathode point, while other thermal electrons will ha\'e wandered farther from this path; conse- quently, the current density in the region of the o--electron is expected to be a reasonable average on which beam spreading due to thermal \elocities may be based. With this understanding of how a is to be cal- culated, we can proceed to the calculation of non-thermal electron trajectories as suggested in item (1). 394 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 The non-thermal paths remain essentially laminar, and with r^ de- noting the radial coordinate of the outermost non-thermal electron, we will make little error in assuming that the current density of non-ther- mal electrons is constant for r < Ve . Consequently, if equal numbers of thermal electrons are assumed to be normally distributed about the cor- responding non-thermal paths, the longitudinal current density as a function of radius can be found in a straightforward way by using (18). The result is J ^ ^_(..,,..) n" R ^-(«^/2.^)^^ frR\ ^ /R\ ^23) Jd Jo a \a^/ \(t/ where /o is the zero order modified Bessel function and the total current is Id = TTVe Jd ' Equation (23) was integrated to give a plot of Jr/Jo versus r/a, with re/a as a parameter and is given as Fig. 6 in Reference 6. It is reproduced here as Fig. 6. Since the only forces acting on elec- trons in the drift region are due to space charge, we may write the equa- tion of motion as where Er is the radial electrical field acting on an electron with radial coordinate r. Since the beam is long and narrow, all electric lines of force may be considered to leave the beam radially so that Er is simpl}^ ob- tained from Gauss' law. Equation (24) therefore becomes -— = --^— / 2irp dr = -— ! — / ■ Iirr dr dt^ zireor Jo Zireor Jo \/2t]V a. (25) 2irenr Jo 27reor From (23) we note that the fraction of the total current within any radius depends only on fe/o- and j'/ct: :il dr ^ / J0')2irr ar / xo ,/o r = - = H-) f '- r.J(r)2.rdr ^''''° (2«) ' Jo ■•r I a C '^dV^]^Fr-j- \(X a t \ BEAM FORMATION WITH ELECTRON GUNS 395 Fig. 6 — Curves showing the current density variation with radius in a beam I which has been dispersed by thermal velocities. Here r« is the nominal beam radius, I r is the radius variable, and <t is the standard deviation defined in equation 17. A family of curves with this ratio, Fr , as parameter has been reproduced : from the Hines-Cutler paper and appears here as Fig. 7. Using this no- tation, (25) becomes dV ^ Vr,/{2V.) j^ Fr di^ 27reo r or d r dz^ jn_ lo Fr^ Fr 27r€0 (27,7a)3/2 J. J. (27) where we have made use of the dc electron drift velocity to make dis- tance the independent variable instead of time, and have defined a quantity K which is proportional to gun perveance. We can now apply (27) to the motion of both the outer (edge) non-thermal electron and the cr-electron. From (26) we see that Fr, and Fg depend only on re/a] 396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 12 11 10 a LLI < u B 1 z z o < o ^ /^ ,/;^ ^ // P> ^ Fr = 0.995/ ^ ^ /^ ^ ^ ^ /^ / rz ^ / >> / ^ z:^ ^ :^ / y. ^ /y 'A %: ^ ;^ ^ Xy '^. ^^ ^^ ^^ w i^ /^ oao^ ^ 1^ ^- oo^ = ^^ 10 re/0- Fig. 7 — Curves showing the fraction, Fr , of the total beam current to be found within any given radius in a beam dispersed by thermal velocities as in Fig. 6. consequently the continuous solution for r^ and r„ (= a) as one moves axially along the drifting beam involves the simultaneous solution of two equations : (fve d~a d^ KFr./re KFJa (28) BEAM FORMATION WITH ELECTRON GUNS 397 0.36 0.32 0.28 0.24 0.16 0.12 0.08 0.04 \ \ 1 \ \ \ \ V V. --- — ■ 8 10 12 14 16 I Fig. 8 — A curve showing the effect of a quantity related to the space charge • force (in the drift region) on a thermal electron with standard deviation a. (See 'equation 28.) which are related by the mutual dependence of Fr^ and Fa on re/a. F„ and Frjve are plotted in Figs. 8 and 9. We may summarize the treatment of the drift region, then, as follows: 1 (a) The input values of r^ and rgJ at the entrance to the anode lens jare obtained from the Pierce gun parameters r^ and 6, while the value of a and aJ at the lens entrance can be obtained as mentioned above by integrating (15) from the cathode, where Mc = and (dfx/dt)c = 1, to the anode plane. (The minus subscripts on r' and a' indicate that these slopes are being evaluated on the gun side of the lens; a plus sub- script will be used to indicate evaluation on the drift region side of the lens.) The values of Ve and a on leaving the lens will of course be their entrance values in the drift region, and the effect of the lens on r/ and a' is simply found in terms of the anode lens correction factor T by use of (20). The value of a at the anode can be obtained from (17) if n is known there. In this regard, (15) can be integrated once to give = 1_/M dt " " r\dt)c{r,/r,y (29) 398 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 LL 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 . — — ■-^ \ X ^ \ I / \ \ J \ / / A 7 \ / \ / \ \ \ \ \ \ \ \ f.,/(reA) \ \ \ s ^, > "-. '"-.. ^-*^^ ■•—.^ 1/ 1 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 1? 0.1 6 0.14 0.12 0.10 0.08 0.06 0.04 0.02 6 7 8 9 10 11 12 13 14 Fig. 9 — Showing quantities related to the effect of the space charge force in the drift region on the outermost non-thermal electron. (See equation 28.) i We can now substitute for transit time in terms of distance and Lang- muir's well known potential function/^ —a. The value of this parameter, for the case of spherical cathode-anode geometry in which we are in- terested, depends only on the ratio fe/f which is equal to Vc/rg . (Because of their frerjuent use in gun design, certain functions of —a are included here as Table I.) In terms of —a, then, the potential in the gun region BEAM FORMATION WITH ELECTRON GUNS 399 Fable I Table of Functions of —a Often Used in Electron Gun Design fc/f (-«)2 (- a)V3 (- a)2/3 difc/r) 1.0 0.0000 0.0000 0.0000 0.0000 1.025 0.0006 0.0074 1.05 0.0024 0.0179 0.134 1.075 0.0052 0.0306 0.173 1.10 0.0096 0.0452 0.212 1.392 0.590 1.15 0.0213 0.0768 0.277 1.20 0.0372 0.1114 0.334 1.767 0.716 1.25 0.0571 0.1483 0.385 1.30 0.0809 0.1870 0.432 2.031 0.790 1.35 0.1084 0.2273 0.476 1.40 0.1396 0.2691 0.519 2.243 0.874 1.45 0.1740 0.3117 0.558 1.50 0.2118 0.3553 0.596 2.423 0.886 1.60 0.2968 0.4450 0.667 2.583 0.915 1.70 0.394 0.5374 0.733 2.725 0.939 1.80 0.502 0.6316 0.795 2.855 0.954 1.90 0.621 0.7279 0.853 2.975 0.970 2.00 0.750 0.8255 0.908 3.087 0.982 2.10 0.888 0.9239 0.961 3.192 0.993 2.20 1.036 1.024 1.012 3.292 1.003 2.30 1.193 1.125 1.061 3.388 1.012 2.40 1.358 1.226 1.107 3.481 1.020 2.50 1.531 1.328 1.152 3.570 1.028 2.60 1.712 1.431 1.196 3.655 1.034 2.70 1.901 1.535 1.239 3.738 1.039 2.80 2.098 1.639 1.280 3.817 1.044 2.90 2.302 1.743 1.320 3.894 1.048 3.00 2.512 1.848 1.359 3.968 1.052 3.1 2.729 1.953 1.397 4.040 1.056 3.2 2.954 2.059 1.435 4.111 1.059 3.3 3.185 2.164 1.471 4.180 1.062 3.4 3.421 2.270 1.507 4.247 1.064 3.5 3.664 2.376 1.541 4.315 1.066 3.6 3.913 2.483 1.576 4.377 1.068 3.7 4.168 2.590 1.609 4.441 1.070 3.8 4.429 2.697 1.642 4.501 1.072 3.9 4.696 2.804 1.674 4.563 1.074 4.0 4.968 2.912 1.706 4.621 1.076 400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 may be written df df i-aaY'^ dt 'V2nV a/2;^ {-a) 2/3 (30) (31) (32) so that upon substitution from (29) and (31), (17) becomes Fig. 5, which has been referred to above, shows O-a . /2eVa 'fcV 'kf- as a function of {fc/fa) as obtained from (32), and allows o-„ to be de- termined easily. Using (20), the value of re+' is given by / Tea , F -,.= -^^_,. = ,/_g-l) (33) where dg is the half-angle of the cathode (and hence the initial angle which the path of a non-thermal edge electron makes with the axis). We may write for 1/Fd 1 V fe /d(-aY"\ Fo 4F 4(-aa)^/VV\rf(fc/r-) 7a (34) In Fig. 10 we plot —falFr, as a function of fjfa for easy evaluation of re+' in (33). Taking the first derivative of (32) with respect to ^, we ob- tain an expression for aJ. Using this in conjunction with (20) and (34) we find 0-+ = Y (r<^i + C2) I (35) where cira d{fc/f) /3 and ^-i/f. ft -(-''"/ (-a)2/3_ ! Ci and C2 are plotted as functions of fc/fa in Fig. 11. (b) After choosing a specific value for r and evaluating K = rj/c/ . BEAM FORMATION WITH ELECTRON GUNS 401 Q LL lU I.O 1.4 1.2 1.0 0.8 0.6 0.4 0.2 \ \ V \ \ ~~~- ■--- 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 40 rcAa Fig. 10 — Curve used in finding ?•«+', the direction of a nonthermal edge elec- tron as it enters the drift region. (See equation 33.) (27r€o(277 Fa) ''), (28) is integrated numerically using the BTL analog com- puter to obtain a and r^ as functions of axial distance along the beam, (c) Knowing a and Ve , other beam parameters such as current dis- tribution and the radius of the circle which would encompass a given percentage of the total current can be found from Figs. 6 and 7. X tvi U 20 15 10 5 -5 -10 -15 -20 -25 -30 POLYNOMIAL REPRESENTATION (ACCURATE WITHIN 2°/o) -OR c, & C2 ,'''' C, = 4.13 fc/ra + 2.67 C2 = 0.635(r^/faf-13.56 rc/fa + 19-33 , ,-' .' ' .-''' .'-' ,'-' \ ^^ -' ^^-' < ^v ■^ X ,.-' ^** ,^-' ''H '^ "^ ^ ^ ^^ \> ^ 20 18 16 14 12 rO O 10 X (J 8 2 10 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 tc/fa Fig. 11 — Curves used in evaluating o-+', the slope of the trajectory of a thermal electron with standard deviation a as it enters the drift region. (See equation 35.) 402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 5. NUMERICAL DATA FOR ELECTRON GUN AND BEAM DESIGN A. Choice of Variables Except for a scaling parameter, the electrical characteristics of an ideal Pierce electron gun are completely determined when three param- eters are specified, e.g., fc/fa , perveance, and Va/T. Also, for the simp- lest case r is equal to 1 so that (since K depends only on gun perveance) in this case no additional parameter is needed. This implies that nor- malized values of ?-/, a, a', and K at the drift side of the anode lens are not independent. If, however, the value of F at the anode lens is taken as an additional variable, four parameters plus simple scaling are re- quired before complete predictions of beam characteristics can be made. In assembling analog computer data which would adequately cover values of fc/fa , perveance, and Va/T which are likely to be of interest to us in designing future guns, we chose to present the major part of our data with T fixed at 1.1. This has seemed to be a rather typical value for r, and by choosing a specific value we decrease the total number of significant variables from 4 to 3. (The effect of variations in T on the minimum radius which contains 95 per cent of the beam is, however, included in Fig. 16 for particular values of Va/T and perveance.) Al- though the boundary conditions for our mathematical description of the beam in a drift space are simplest when expressed in terms of Vg , r/, a and ct', we have attempted to make the results more usable by express- ing all derived parameters in terms of fc/fa , s/Va/T, and the perveance, P. B. Tabular Data The rather extensive data obtained from the analog computer for the r = 1.1 case and for practical ranges in perveance, Ve/T, and fc/fa are summarized in Tables IIA to E where the parameters r^ and a which specify the beam cross section are given as functions of axial distance from the anode plane. Some feeling for the decrease in accuracy to be expected as the distance from the anode plane increases can be obtained by reference to Section 6B where experiment and theory are compared over a range of this axial distance parameter. C. Graphical Data, Including Design Charts and Beam Profdes In typical cases, the designer of Pierce electron guns is much more concerned with the beam radius at the axial position where it is smallest (and in the axial position of this minimum) than he is in the general BEAM FORMATION WITH ELECTRON GUNS 403 jspreadiug of the beam with distance. This is true because, in microwave beam tubes, the beam from a magnetically shielded Pierce gun normally enters a strong axial magnetic field near a point where the radius is a minimum, so that magnetic focusing forces largely determine the beam's subsequent behavior. The analog computer data has therefore been re- processed to stress the dependence of the beam's minimum diameter and the corresponding axial position of the minimum on the basic design iparameters fdfa , perveance, and s/Va/T. As a first step in this direc- tion, the radius, rgs , of a circle which includes 95 per cent of the beam : I current is obtained as a function of axial position along the beam. Such idata are shown graphically in Fig. 12. Finally, the curves of Fig. 12 are . lused in conjunction with the tabular data to obtain the "Design Curves" of Fig. 13 where all of the pertinent information relating to the beam at its minimum diameter is presented. \D. Example of Gun Design Using Design Charts Assume that we desire an electron gun with the following properties : anode voltage Va = 1,080 volts, cathode current Ip = 7.1 ma, and mini- mum beam diameter 2(r95)min = 0.015 inches. Let us further assume a cathode temperature T = 1080° Kelvin, an available cathode emission density of 190 ma per square cm, and an anode lens correction factor of r = 1.1. From these data we find -x/Va/T = 1.0, perveance P = 0.2 X 10"^ amps/(volts)^''" and (r95)min/''c = 0.174. Reference to the de- sign chart, Fig. 13, now gives us the proper value for fc/fa : using the upper set of curves in the column for y/Va/T =1.0 we note the point of intersection between the horizontal line for {rgr^^i^/rc = 0.174 and the perveance line P = 0.2, and read the value of fc/fa (= 2.8) as the corresponding abscissa. The convergence angle of the gun, de , is now simply determined fi'om the equation^^ de = cos-^ {\ - t|^ X 10^) (37) {Qe is found to be 13.7° in this example) and the potential distribution in the region of the cathode can be obtained from (30). When this point has been reached, the gun design is complete except for the shapes of the beam forming electrode and the anode, which are determined with the aid of an electrolytic tank in the usual way. The radius of the anode hole which will give a specified transmission can be found by obtaining (re/a)a through the use of Fig. 5, and then choosing the anode radius from Fig. 7. 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'S ^ 1 ii f 10/ / 1 "v V ^// r ^0 y # \s \\ \ \; \ \ V \ > A. N \ ^ N M^ vl '^i i ^ ); i ^ ^ t? N <D V "^sVs \ \ \ \ s 1 ^ i A <o\ A /' r \ ». \. ^ .\ \ <s ^ \^ \ ^ K 1 \ ^ ii I J m ii7 ^ y ti <D ID d *^ V \ \ \ I II w \ ^ \ V > ^ II 01 lO oil fVi, ' m ^ (\l >? t>J o CO 6 d 5_0l X SOO=d o o (\J 00 '- _0l X 10 = d 410 \ \ \ x^ \ N ^^ \ ""''^ '^ \ V/ ^ ^ U ^ / V \, -^ \ \ \ l1 Vv "^^^^\" fe o> = ^«^^-"~. '^^ k ^ ^ \ \ \ ^\ t^ .-^ \- °^ ^ ^ \ / ^ \ W ^ V \v. "0^ '■^ 3 k^ V ^ ^ 0-^ Ij^ M^ Jj w ^ ^ \J ^ ^r^^ ^ \ vs; ^. \ "A <r)l 1 "^'/ t-V \V \ A k '<'> o <\J CO ^ o o o o (\J OO d d CO &-; Q t- CO > T) fl 'J c3 N I5~ 1^ o (\J ST ^^ — ' a) a 0) > Lh «1) OJ a a • p— 1 r^ CO o -t^ rt Si -il ^ !> ^ o %-. OJ o ^3 <D O <A C 3 ^ tJU OJ o ;_ 5<-l O CM a; N ^ s^ ^ ^ (O o '^ 9_0l X 2-0 = d 9.01 X fO = d C CO CO a; > a o <D >o CO .2 O -C CO CO > 3 o fci) • r-t fa 411 q > > \ \ 1 \ \ ^ \ . ^-- t^ ^ J^ IJ?£!\ I ^ tsl > V \ \ \ \ \"^ ^ L ^^ \ ^ ^V. K ^ C> ■- > ^ •i \& ^ ^N K\ ■^ ^ d II 1— ^ ^ V s. > ^ ^ 1^ \ y ~A^ K ^ > o N CO 9_01X 90:=c) 412 d |5- o d > C3 O • i-t 03 c3 > o OS c a faC o u 0) o o o3 3 CO > C o o3 o3 O -a EC a> > O O .— I bb BEAM FORMATION WITH ELECTRON GUNS 413 we find less than 1 per cent anode interception if anode hole radius = 0.93 r^a + 2o-a (38) Additional information about the axial position of (r95)min and the cur- rent density distribution in the corresponding transverse plane is con- tained in Fig. 13. The second set of curves in the \/Va/T = 1 column gives Zm\n/Tc — 2.42 for this example, so that we would predict Zmin = distance from anode to (r95)inin = 0.104'' The remaining 3''^ and 4*^^ sets of curves in the ■\/Va/T = 1 column allow us to find o- and re/a- at ^min . In particular we obtain a = 0.0029" and I'e/o = 0.8, and use Fig. 6 to give the current density distribution at 2min .* Section VI contains experimental data which indicate a some- what larger value for 2m in than that obtained here. However the pa- rameter of greatest importance, (r95)niin , is predicted with embarrassing precision. For those cases in which additional information is required about the beam shape at axial points other than ZnVin , the curves of Fig. 12 or the data of Table II may be used. 6. COMPARISON OF THEORY WITH EXPERIMENT In order to check the general suitability of the foregoing theory and the usefulness of the design charts obtained, several scaled-up versions of Pierce type electron guns, including the gun described in Section 5D, were assembled and placed in the double-aperture beam analyzer de- scribed in Reference 7. A. Measurement of Current Densities in the Beam Measurements of the current density distributions in several trans- verse planes near Smin were easily obtained with the aid of the beam analyzer. The resulting curve of relative current density versus radius at the experimental 2min is given in Fig. 14 for the gun of Section 52). (This curve is further discussed in Part C below.) For this case, as well as for all others, special precautions were taken to see that the gun was functioning properly : In addition to careful measurement of the size and position of all gun parts, these included the determination that the dis- tribution of transverse velocities at the center of the beam was smooth * When j'c/o- < 0.5, the current density distribution depends almost entirely on a, and, in only a minor way, on the ratio Te/a- so that in such cases this ratio need not be accurately known. q / / / / / / / A y V / i / ^ / y y / /a ^ ^ ^ 4 <^^ 1 1 1 / '/ // 1 / YfA/. h (^ Y // ^*^.. 00 <o in 'i- ro (\j — (D <o in ^ n ry o O o o 6 O d o o d o d o d o d o o o d ^J/''(N'^J) Dj/Nm^ o m IS-. (\j C\J o — II // / / / / o/ ox o_ /^ A i^ ^ / c J A ^ / i i ^ s^ ^ 1 1 1 O 00 <D in -■ d do ro d rv; d /, // // y <• ^ // \7/ i / <. /: ^ A c ,(. cc ^(f ^>:-5> ^^~5 ~^^^^r if'i^ — CO ID m ^ do o o o d d d d p n It] it- in IS- J d II <!. LU LU Q mto ^, xz a Oo LU tr > u. o inlnj O > n / / V/ 1 2 < / / // // 1 / / A / / 1 D X 1 "^ 1 O / \ ^ 4 / ^ '^ ^ ^ /■ -f 1 1 1 3 CO ID in '3- (O (\J - d O O O o o / A / / // A y oy 1 / / fo Y / \ / ^^-^ ~~, 0j/S6(N,^j) - <0 o ^ i It'-' 2 m 11-. ' 4 « Tj- fO (M o- 414 / 1, V/ ' / / / // A / o 1 d /d ( \ V u 1 V \ \^ ^^ \ ll;.:: o CM d d d NIIAI iV CO o d o d o <o ^■j./d O 00 (O \ \«1 1i> 1 J CM ^^ J f 00 6 'J- d o \ /§ ' If) (D O — d O CO <o Nl^^l XV d/^j. q <o It, 1 I III ijiii ii^ f ^ _,^^ ^ ^ 00 d _ 1 > ^ d /d is«p o CO (D lU •^ lii^ ry o NIW IV in ^ fO N I IN IV i?/^J fvi 415 416 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 12 11 10' z UJ o 5 UJ > ^ \PREDICTED ^■o-.. -IN \ measured"' ^ V 4 \ ^. N V \ k \ "V. \ \ } ^ ^.. P" ^-Gr.^ ==^ 01 23456789 RADIUS IN MILS Fig. 14 — Current density distribution in a transverse plane located where the 95 per cent radius is a minimum. The predicted and measured curves are normal- ized to contain the same total current. (The corresponding prediction from the universal beam spread curve would show a step function with a constant relative current density of 64.2 for r < 1.2 mils and zero beyond.) The gun parameters are given in Section 5D. and generally Gaussian in form, thereby indicating uniform cathode emission and proper boundary conditions at the edge of the beam near the cathode. The ejffect of positive ions on the beam shape was in every I case reduced to negligible proportions, either by using special pulse techniques, or by applying a small voltage gradient along the axis of the beam. B. Comparison of the Experiinentally Measured Spreading of a Beam with that Predicted Theoretically From the experimentally obtained plots of current density versus radius at several axial positions along the beam, we have obtained at each position (by integrating to find the total current within any radius) a value for the radius, rgs , of that circle which encompasses 95 per cent of the beam. For brevity, we call the resulting plots of rgs versus axial distance, "beam profiles". The experimental profile for the giui de- scribed in Section 5D is shown as curve A in Fig. 15(a). Curve B shows the profile as predicted by the methods of this paper and obtained from Fig. 12. Curve C is the corresponding profile which one obtains by the Hines-Cutler method, and Curve D represents Tq^ as obtained from the BEAM FORMATION WITH ELECTRON GUNS 417 CO 20 18 16 14 12 t- 8 2 I 50 45 40 35 if) 30 Z 25 l? 20 15 10 (a) GUN PARAMETERS: fc/fa=2.8 s / ^ \, (C)j / 1 / e = i3.7° VVa/T-i.o \ ^> k / / / [B]/ r rc = 0.043" (A) EXPERIMENT (B) METHODS OF THIS PAPER (C) HINES-CUTLER METHOD (D) UNIVERSAL BEAM SPREAD CURVE \ ^^ V / / / / / \ N s. <; >^ ^ '4 / ^ <. \ \, "^ ^>3e \ \, y /(D) \ \ y y "~~- ^^ ^ 40 80 120 160 200 240 Z, DISTANCE FROM IDEAL ANODE IN MILS 280 320 (b) i /(C) y GUN PARAMETERS: f c/fa = 2.5 1 1 1 / / e = 8° 1.0 / 1 * y ^B) ^/V, /T- \ x^ V a/ rc = 0.150" / / / f y /^ V ^ V ^ ^***^^ •■ • • } \ X X "^ , -» -^ <^ — ■^ (A) \ ^^ .^ y ^-- ^^ ^ (D) 100 200 300 400 500 600 Z, DISTANCE FROM IDEAL ANODE IN MILS 700 800 Fig. 15 — Beam profiles (using an anode lens correction of r = 1.1 and the gun parameters indicated) as obtained (A) from experiment, (B) bj^ the methods of this paper, (C) Hines-Cutler method, (D) by use of the universal beam spread curve. universal l^eam spread curve'" (i.e., under the assumption of laminar flow and gradual variations of beam radius with distance) . Note that in each case a value of 1.1 has been used for the correction factor, r, repre- senting the excess divergence of the anode lens. The agreement in (/'95)min as obtaiucd from Curves A and B is remarkably good, but the axial position of (r95)min in Curve A definitely lies beyond the correspond- 418 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ing inininumi position in Curve B. Fortunately, in the gun design stage, one is usually more concerned with the value of (r95)min than with its exact axial location. The principal need for knowing the axial location of the minimum is to enable the axial magnetic field to build up suddenly in this neighborhood. However, since this field is normally adjusted ex- perimentally to produce best focusing, an approximate knowledge of 2m in is usually adequate. In Fig. 15b we show a similar set of experimental and theoretical beam profiles for another gun. The relative profiles are much the same as in Fig 15a, and all of several other guns measured yield experimental points similarly situated with respect to curves of Type B. C. Comparison of Experimental and Theoretical Current Density Dis- tributions where the Minimum Beam Diameter is Reached In Fig. 14 we have plotted the current density distribution we would have predicted in a transverse plane at ^min for the example introduced in Section 5Z). Here the experimental and theoretical curves are nor- malized to include the same total currents in their respective beams. The noticeable difference in predicted and measured current densities at the center of the beam does not appreciably alter the properties such a beam would have on entering a magnetic field because so little total current is actually represented by this central peak. D. Variation of Beam Profile with T All of the design charts have been based on a value of T = 1.1, which is typical of the values obtained by the methods of Section 3. When appreciably different values of F are appropriate, we can get some feel- ing for the errors involved, in using curves based on T = 1.1, by refer- ence to Fig. 16. Here we show beam profiles as obtained by the methods of this paper for three values of F. The calculations are again based on the gun of Section 5D, and a value of just over 1.1 for F gives the ex- perimentally obtained value for (r95)min . 7. SOME ADDITIONAL REMARKS ON GUN DESIGN In previous sections we have not differentiated between the voltage on the accelerating anode of the gun and the final beam voltage. It is important, howovei', that the separate functions of these two voltages be kept clearly in mind: The accelerating anode determines the total current drawn and largely controls the shaping of the beam; the final beam voltage is, on the other hand, chosen to give maximum interaction between the electron beam and the electromagnetic waves traveling along the slow wave circuit. As a consequence of this separation of func- , BEAM FORMATION WITH ELECTRON GUNS 419 0.006 0.02 0.18 0.20 0.22 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Z, DISTANCE FROM IDEAL ANODE IN INCHES Fig. 16 — Beam profiles as obtained by the methods of this paper for the gun parameters given in Section bD. Curves are shown for three values of the anode lens correction, viz. T = 1.0, 1.1, and 1.2. tions, it is fouiicl that some beams which are difficult or impossible to obtain with a single Pierce-gun acceleration to final beam voltage may be obtained more easily by using a lower voltage on the gun anode. The acceleration to final beam voltage is then accomplished after the beam has entered a region of axial magnetic field. Suppose, for example, that one wishes to produce a 2-ma, 4-kv beam with (rgs/rc) = 0.25. If the cathode temperature is 1000°K, and the gun anode is placed at a final beam voltage of 4 kv, we have \^Va/T = 2 and P = 0.008. From the top set of curves under \^Va/T = 2 in Fig. 13, we find (by using a fairly crude extrapolation from the curves shown) that a ratio of fc/fa'^ 3.5 is required to produce such a beam. The value of {ve/o-) at Zmin IS therefore less than about 0.2 so that there is little x'mblance of laminar flow here. On the other hand we might choose r, = 250 volts so that a/fT^ = 0.5 and P = 0.51. From Fig. 13* we than obtain fc/fa = 2.6 and (re/o-)min = 0.8 for the same ratio of '■'joAc(= 0.25). While the flow could still hardly be called laminar, it is (•(jnsiderably more ordered than in the preceding case. Here we have in- cluded no correction for the (convergent) lens effect associated with the post-anode acceleration to the final beam voltage, F = 4 kv. Calculations of the Hines-Cutler type will always predict, for a given set of gun parameters and a specified anode lens correction, a minimum beam size which is larger than that predicted by the methods of this ])aper. Nevertheless, in many cases the difference between the minimum sizes predicted by the two theories is negligible so long as the same anode lens correction is used. The extent to which the two theories agree ob- 420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 viously depends on the magnitude of Velo. When rel(T as calculated by the Hines-Cutler method (with a lens correction added) remains greater than about 2 throughout the range of interest, the difference between the corresponding values obtained for rgs will be only a few per cent. For these cases where rja does not get too small, the principal advan- tages of this paper are in the inclusion of a correction to the anode lens formula and in the comparative ease with which design parameters may be obtained. In other cases r^la may become less than 1, and the theory presented in this paper has extended the basic Hines-Cutler approach so that one may make realistic predictions even under these less ideal conditions where the departure from a laminar-type flow is quite severe. ACKNOWLEDGMENT We wish to thank members of the Mathematical Department at B.T.L., particularly H. T. O'Neil and Mrs. L. R. Lee, for their help in programming the problem on the analog computer and in obtaining the large amount of computer data involved. In addition, we wish to thank J. C. Irwin for his help in the electrolytic tank work and both Mr. Irwin and W. A. L. Warne for their work on the beam analyzer. REFERENCES 1. Pierce, J. R., Rectilinear Flow in Beams, J. App. Phys., 11, pp. 548-554, Aug., 1940. 2. Samuel, A. L., Some Notes on the Design of Electron Guns, Proc. I.R.E., 33, pp. 233-241, April, 1945. 3. Field, L. M., High Current Electron Guns, Rev. Mod. Phys., 18, pp. 353-361, July, 1946. 4. Davisson, C. J., and Calbick, C. J., Electron Lenses, Phys. Rev., 42, p. 580, Nov., 1932. 5. Helm, R., Spangenburg, K., and Field, L. M., Cathode-Design Procedure for Electron Beam Tubes, Elec. Coram., 24, pp. 101-107, March, 1947. 6. Cutler, C. C, and Hines, M. E., Thermal Velocity Effects in Electron Guns, Proc. I.R.E., 43, pp. 307-314, March, 1955. 7. Cutler, C. C, and Saloom, J. A., Pin-hole Camera Investigation of Electron Beams, Proc. I.R.E., 43, pp. 299-306, March, 1955. 8. Hines, M. E., Manuscript in preparation. 9. Private communication. 10. See for example, Zworykin, V. K., et al.. Electron Optics and the Electron Microscope, Chapter 13, Wiley and Sons, 1945, or Klemperer, O., Electron Optics, Chapter 4, Cambridge Univ. Press, 1953. 11. Brown, K. L., and Siisskind, C., The Effect of the Anode Aperature on Po- tential Distribution in a "Pierce" Electron Gun, Proc. I.R.E., 42, p. 598, March, 1954. 12. See, for example, Pierce, J. R., Theory and Design of Electron Beams, p. 147, Van Nostrand Co., 1949. 13. See Reference 6, p. 5. 14. Langmuir, I. L., and Blodgett, K., Currents Limited by Space Charge Be- tween Concentric Spheres, Phys. Rev., 24, p. 53, July, 1924. 15. See Reference 12, p. 177. 16. See Reference 12, Chap. X. Theories for Toll Traffic Engineering in the U.S.A.* By ROGER I. WILKINSON (Manuscript received June 2, 1955) Present toll trunk traffic engineering practices in the United States are reviewed, and various congestion formulas compared with data obtained on long distance traffic. Customer habits upon meeting busy channels are noted and a theory developed describing the probable result of permitting subscribers to have direct dialing access to high delay toll trunk groups. Continent-wide automatic alternate routing plans are described briefly, in which near no-delay service will permit direct customer dialing. The presence of non-random overflow traffic from high usage groups co7nplicates the estimation of correct quantities of alternate paths. Present methods of solving graded multiple problems are reviewed and found unadaptable to the variety of trunking arrangements occurring in the toll plan. Evidence is given that the principal fluctuation characteristics of overflow- type of non-random traffic are described by their mean and variance. An approximate probability distribution of simultaneous calls for this kind of non-random traffic is developed, and found to agree satisfactorily with theo- retical overflow distributions and those seen in traffic simidations. A method is devised using ^^ equivalent random''^ traffic, which has good loss predictive ability under the "lost calls cleared" assumption, for a diverse field of alternate route trunking arrangements. Loss comparisons are made with traffic simulation residts and with observations in exchanges. Working curves are presented by which midti-alternate route trunking systems can be laid out to meet economic and grade of service criteria. Exam- ples of their application are given. Table of Contents 1 . Introduction 422 2. Present Toll Traffic Engineering Practice 423 * Presented at the First International Congress on the Application of the Theory of Probability in Telephone Engineering and Administration, Copen- hagen, June 21, 1955. 421 422 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 3. Customers Dialing on Groups with Considerable Delay 431 3.1. Comparison of Some Formulas for Estimating Customers' NC Service on Congested Groups 434 4. Service Requirements for Direct Distance Dialing by Customers 436 5. Economics of Toll Alternate Routing 437 6. New Problems in the Engineering and Administration of Intertoll Groups Resulting from Alternate Routing 441 7. Load-Service Relationships in Alternate Route Systems 442 7.1. The "Peaked" Character of Overflow Traffic 443 7.2. Approximate Description of the Character of Overflow Traffic 446 7.2.1. A Probability Distribution for Overflow Traffic 452 7.2.2. A Probability Distribution for Combined Overflow Traffic Loads 457 7.3. Equivalent Random Theory for Prediction of Amount of Traffic Over- flowing a Single Stage Alternate Route, and Its Character, with Lost Calls Cleared 461 7.3.L Throwdown Comparisons with Equivalent Random Theory on Simple Alternate Routing Arrangements with Lost Calls Cleared 468 7.3.2. Comparison of Equivalent Random Theory with Field Results on Simple Alternate Routing Arrangements 470 7.4. Prediction of Traffic Passing Through a Multi-Stage Alternate Route Network 475 7.4.1. Correlation of Loss with Peakedness of Components of Non- Random Offered Traffic 481 7.5. Expected Loss on First Routed Traffic Offered to Final Route 482 7.6. Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped Alternate Route 486 8. Practical Methods for Alternate Route Engineering 487 8.1. Determination of Final Group Size with First Routed Traffic Offered Directly to Final Group 490 8.2. Provision of Trunks Individual to First Routed Traffic to Equalize Service 491 8.3. Area in Which Significant Savings in Final Route Trunks are Real- ized by Allowing for the Preferred Service Given a First Routed Traffic Parcel 494 8.4. Character of Traffic Carried on Non-Final Routes 495 8.5. Solution of a Typical Toll Multi-Alternate Route Trunking Arrange- ment : Bloomsburg, Pa 500 9. Conclusion 505 Acknowledgements 506 References 506 Abridged Bibliography of Articles on Toll Alternate Routing 507 Appendix I: Derivation of Moments of Overflow Traffic 507 Appendix II: Character of Overflow when Non-Random Traffic is Offered to a group of Trunks 511 1. INTRODUCTION It has long been the stated aim of the Bell System to make it easily and economically possible for any telephone customer in the United States to reach any other telephone in the world. The principal effort in this direction by the American Telephone and Telegraph Company and its associated operating companies is, of course, confined to inter- connecting the telephones in the United States, and to providing com- munication channels between North America and the other countries of the world. Since the United States is some 1500 miles from north to fSOuth and 3000 miles from east to west, to realize even the aim of fast THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 423 and economical service between customers is a problem of great magni- tude; it has engaged our planning engineers for many years. There are now 52 million telephones in the United States, over 80 per cent of which are equipped with dials. Until quite recently most telephone users were limited in their direct dialing to the local or immediately sur- rounding areas and long distance operators were obliged to build up a circuit with the aid of a "through" operator at each switching point. Both speed and economy dictated the automatic build-up of long toll circuits without the intervention of more than the originating toll oper- ator. The development of the No. 4-type toll crossbar switching system with its ability to accept, translate, and pass on the necessary digits (or lujuivalent information) to the distant office made this method of oper- ation possible and feasible. It was introduced during World War II, and now by means of it and allied equipment, 55 per cent of all long distance calls (over 25 miles) are completed by the originating operator. As more elaborate switching and charge-recording arrangements were developed, particularly in metropolitan areas, the distances which cus- tomers themselves might dial measurably increased. This expansion of the local dialing area was found to be both economical and pleasing to the users. It was then not too great an effort to visualize customers dialing to all other telephones in the United States and neighboring countries, and perhaps ultimately across the sea. The physical accomplishment of nationwide direct distance dialing which is now gradually being introduced has involved, as may well be imagined, an immense amount of advance study and fundamental plan- ning. Adequate transmission and signalling with up to eight intertoll trunks in tandem, a nationwide uniform numbering plan simple enough to be used accurately and easily by the ordinary telephone caller, pro- ^ ision for automatic recording of who called whom and how long he talked, with subsequent automatic message accounting, are a few of man}^ problems which have required solution. How they are being met is a romantic story beyond the scope of the present paper. The references given in the bibliography at the end contain much of the history as well as the plans for the future. • 2. PRESENT TOLL TRAFFIC ENGINEERING PRACTICE There are today approximately 116,000 intertoll trunks (over 25 miles in length) in the Bell System, apportioned among some 13,000 trunk groups. A small segment of the 2,600 toll centers which they interconnect is shown in Fig. 1. Most of these intertoll groups are presently traffic engineered to operate according to one of several so-called T-schedules: T-8, T-15, T-30, T-60, or T-120. The number following T (T for Toll) is 424 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 KEY O TOLL CENTERS INTERTOLL TRUNK GROUPS Fig. 1 — Principal intertoll trunk groups in Minnesota and Wisconsin. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 425 4 5 6 7 8 9 10 NUMBER OF TRUNKS 30 40 50 Fig. 2 — Permitted intertoU trunk occupancy for a 6.5-minute usage time per message. the expected, or average, delay in seconds for calls to obtain an idle trunk in that group during the average Busy Season Busy Hour. In 1954 the system "average trunk speed" was approximately 30 seconds, re- sulting from operating the majority of the groups at a busy-hour trunk- ling efficiency of 75 to 85 per cent in the busy season. The T-engineering tables show permissible call minutes of use for a 426 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 wide range of group sizes, and several selections of message holding times. They were constructed following summarization of many obser- vations of load and resultant average delays on ringdown (non-dial) intertoll trunks.^ Fig. 2 shows the permissible occupancy (efficiency) of various trunk group sizes for 6.5 minutes of use per message, for a va- riety of T-schedules. It is perhaps of somfe interest that the best fitting curves relating average delay and load were found to be the well-known Pollaczek-Crommelin delay curves for constant holding time — this in spite of the fact that the circuit holding times were far indeed from having a constant value. A second, and probably not uncorrected, observation was that the per cent "No-Circuit" (NC) reported on the operators' tickets showed consistently lower values than were measured on group-busy timing de- vices. Although not thoroughly documented, this disparity has generally been attributed to the reluctance of an operator to admit immediately the presence of an NC condition. She exhibits a certain tolerance (very difficult to measure) before actually recording a delay which would recjuire her to adopt a prescribed procedure for the subsequent handling of the call.* There are then two measures of the No-Circuit condition which are of some interest, the "NC encountered" by operators, and the "NC existing" as measured by timing devices. It has long been observed that the distribution of numbers n of simul- taneous calls found on T-engineered ringdown intertoll groups is in re- markable agreement with the individual probability terms of the Erlang "lost calls" formula, f n — a ' a e fin) = ^-^^ (1) e E- n=o n! where c = number of paths in the group, a' = an enhanced average load submitted such that a'[l — Ei^c(a')] = L, the actual load carried, and Ei^cid') = fie) = Erlang loss probability (commonly called Er- lang B in America). An example of the agreement of observations with (1) is shown in Fig. 3, where the results of switch counts made some years ago on many ringdown circuit groups of size 3 are summarized. A wide range of "sub- * Upon finding No-Circuit, an operator is instructed to try again in 30 seconds and GO seconds (before giving an NC report to the customer), followed by addi- tional attempts 5 minutes and 10 minutes later if necessary. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 427 0.10 0.2 0.5 1.0 2 AVERAGE "submitted" LOAD IN ERLANIGS Fig. 3 — Distributions of simultaneous calls on three-trunk toll groups at .\lbany and Buffalo. I nit ted" loads a' to produce the observed carried loads is required. On Fig. 4 are shown the corresponding comparisons of theory and obser- vations for the proportions of time all paths are busy ("NC Existing") for 2-, 4-, 5-, 7-, and 9-circuit groups. Good agreement has also been ob- served for circuit groups up to 20 trunks. This has been found to be a stable relationship, in spite of the considerable variation in the actual practices in ringdown operation on the resubmission of delayed calls. Since the estimation of traffic loads and the subsequent administration of ringdown toll trunks has been performed principally by means of Group Busy Timers (which cumulate the duration of NC time), the Erlang relationship just described has been of great importance. 428 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 With the recent rapid increase in operator dialed intertoll groups, it might be expected that the above discrepancy between " % NC encoun- tered" and "% NC existing" would disappear — for an operator now initiates each call unaware of the momentary state of the load on any particular intertoll group. By the use of peg count meters (which count calls offered) and overflow call counters, this change has in fact been observed to occiu'. ]\Ioreo^'er, since the initial re-trial intervals are com- monly fairly short (30 seconds) subsequent attempts tend to find some of the previous congestion still existing, so that the ratio of overflow to peg count readings now exceeds slightly the "% NC existing." This situation is illustrated in Fig. 5, which shows data taken on an operator- 1.0 AVERAGE SUBMITTED LOAD Fig. 4 — Observed proportions of time all trunks were busy on Albany and Buffalo groups of 2, 4, 5, 7, and 9 trunks, THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 429 u z o z I- UJ HI 5 ul _i _i < o U- o z o (- cc o a. o tr a. 0.001 12 14 L = LOAD CARRIED IN ERLANGS 18 Fig. 5 — Comparison of NC data on a 16-trunk T-engineered toll group with various load versus NC theories. dialed T-engineered group of 16 trunks between Newark, N. J., and Akron, Ohio. Curve A shows the empirically determined "NC encoun- tered" relationship described above for ringdown operation; Curve B gives the corresponding theoretical "NC existing" values. Lines C and D give the operator-dialing results, for morning and afternoon busy hours. The observed points are now seen generally to be significantly above Curve B.* At the same time as this change in the "NC encountered" was occur- ring, due to the introduction of operator toll dialing, there seems to have l)een little disturbance to the traditional relationship between load * The observed point at 11 erlangs which is clearly far out of agreement with the remainder of the data was produced by a combination of high-trend hours and an hour in which an operator apparently made many re-t^rials in rapid suc- cession. 430 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 u. 10 z o o 5 o m rvj ti _r < (- z LU z o z o i tr UJ I/) § o «--- LIMIT OF OBSERVED DATA i [ oiT / / / / / / / / / / / y / / /' ^•^ /^ ^ ^ ««- ^ ^ Tt^^ ^•^^ ^ s:;^ 8 If) o 0> o (0 (O o If) o in o o in tvj o in SBIONII^ 1- a3AO SidlAjaiiV dO iN3D«3d THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 431 carried and " % NC existing." C. J. Truitt of the A.T. & T. Co. studied i a number of operator-dialed T-engineered groups at Newark, New Jersey, in 1954 with a traffic usage recorder (TUR) and group-busy timers, and found the relationship of equation (1) still good. (This analysis has not been published.) A study by Dr. L. Kosten has provided an estimate of the probability that when an NC condition has been found, it will also appear at a time T later." When this modification is made, the expected load-versus-NC relationship is shown by Curve E on Fig. 5. (The re-trial time here was taken as the operators' nominal 30 seconds; with 150-second circuit-use time the return is 0.2 holding time.) The observed NC's are seen to lie slightly above the E-curve. This could be explained either on the basis that Kosten's analysis is a lower limit, or that the operators did not strictly observe the 30-second return schedule, or, more probably, a combination of both. 3. CUSTOMERS DIALING ON GROUPS WITH CONSIDERABLE DELAY It is not to be expected that customers could generally be persuaded to wait a designated constant or minimum re-trial time on their calls which meet the NC condition. Little actual experience has been accumulated on customers dialing long distance calls on high-delay circuits. However, it is plausible that they would follow the re-trial time distributions of customers making local calls, who encounter paths-busy or line-busy signals (between which they apparently do not usually distinguish). Some information on re-trial times was assembled in 1944 by C. Clos by observing the action of customers who received the busy signal on 1,100 local calls in the City of New York. As seen in Fig. 6, the return times, after meeting "busy," exhibit a marked tendency toward the exponential distribution, after allowance for a minimum interval required for re- dialing. An exponential distribution with average of 250 seconds has been I fitted by eye on Fig. 6, to the earlier ■ — and more critical — customer re- turn times. This may seem an unexpectedly long wait in the light of indi- vidual experience; however it is probably a fair estimate, especially since, following the collection of the above data, it has become common practice for American operating companies in their instructional lit- erature to advise customers receiving the busy signal to "hang up, wait a few minutes, and try again." The mathematical representation of the situation assuming exponen- I tial return times is easily formulated. Let there be .r actual trunks, and 432 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 imagine y waiting positions, whore y is so large that few calls are re- jected.* Assume that the offered load is a erlangs, and that the calls have exponential conversation holding times of unit average duration. Finally \ let the average return time for calls which have advanced to the waiting > positions, be 1/s times that of the unit conversation time. The statistical j equilibrium equation can then be written for the probability j\m, n) (j that m calls are in progress on the x trunks and n calls are waiting on the y storage positions: ■ /(w, n) = aj{m — 1, n) dt + s(w + l)/(m — 1, n + 1) dt ''■) + (m + \)J{m + 1, n) dt + a/(.r, n - 1) dH^ (2) + [1 - (a*** + sn**) dt - m dt]f(m, n) ^ where ^ m ^ .-r, ^ w ^ //, and the special limiting situations are recognized by: ■* Include term only when m — x **■ Omit sn when m = x *** Omit a when m = x and n = y Equation (2) reduces to (a*** + snifif + m)f{m; n) = af{m — 1, n) 1 + s(n + l)/(m - 1, w + 1) (3) + (m + l)/(w + 1, n) + af(x, n - !)•, Solution of (3) is most easily effected for moderate values of x and y by first setting f(x, ?/) = 1 .000000 and solving for all other /(/?? , ?? ) in X y terms of /(o:, ?/). Normahzing through zl 11f(m, n) = 1.0, then gives m=0 n=0 the entire f(m, n) array. The proportion of time "NC exists," will, of course be Z Six, n) (4) n=0 and the load carried is L = Xl X wi/(m, n) (5) The proportion of call attempts meeting NC, including all re-trials * The quant itjr y can also be chosen so that some calls are rejected, thus roughly describing those calls abandoned after the first attempt. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A, 433 will be W{x, a, s) = Expected overflow calls per unit time Expected calls offered per unit time Z (a + sn)jXx, n) - , ./ ^ ^^^ sn -\- af{x, y) n=0 X y S 2 (« + sn)f(m, n) a -{- sn m=0 71=0 X y in which n = ^ 2^ nf(7n, n). And when y is chosen so large that/(.r, y) 7H = 71=0 is negligible, as we shall use it here, L = a W(x, a, s) = sn a -\- sn (5') (6') 1^ 0.5 < "^O 0.4 ilZ Oo ZZ 0.3 Ol- pllJ o5 0.2 Q. o ? 0.1 6 TRUNKS / // APOISSON ' ^1 P(C,L) 5=0.6 2 4 6 8 L=LOAD CARRIED IN ERLANGS APOISSON P(C,L) fly >^- f I6j _, 8 10 12 14 L = LOAD CARRIED IN ERLANGS Fig. 7 — ■ Comparison of trunking formulas. 434 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 I This formula provides a means for estimating the grade of service which customers might he expected to receive if asked to dial their calls over moderate-delay or high-delay trunk groups. For a circuit use length of 150 seconds, and an average return time of 250 seconds (as on Fig. 6), both exponential, the load-versus-proportion-NC curves for 6 and IG trunks are given as curves (3) on Fig. 7. For example with an offered (= carried) load of a = 4.15 erlangs on 6 trunks we should expect to find 27.5 per cent of the total attempts resulting in failure. For comparison with a fixed return time of NC-calls, the IF-formula curves for exponential returns of 30 seconds (s = 5) and 250 seconds (s = 0.6) averages are shown on Fig. 5. The first is far too severe an assumption for operator performance, giving NC's nearly double those actually observed (and those given by theory for a 30-second constant return time). The 250-second average return, however, lies only slightly above the 30-second constant return curve and is in good agreement with the data. Although not logically an adequate formula for interpreting Peg Count and Overflow registrations on T-engineered groups under operator dialing conditions, the IF-formula apparently could be used for this purpose with suitable s-values determined empirically. 3.1. Comparison of Some Formulas for Estimating Customers' NC Service on Congested Groups , 1 As has been previously observed, a large proportion of customers who receive a busy signal, return within a few minutes (on Fig. 6, 75 per cent of the customers returned within 10 minutes). It is well known too, that under adverse service conditions subscriber attempts (to reach a par- ticular distant office for example) tend to produce an inflated estimate of the true offered load. A count of calls carried (or a direct measurement of load carried) will commonly be a closer estimate of the offered load than a count of attempts. An exception may occur when a large propor- tion of attempts is lost, indicating an offered load possibly in excess even of the number of paths provided. Under the latter condition it is diffi- cult to estimate the true offered load by any method, since not all the attempts can be expected to return repeatedly until served; instead, a significant number will be abandoned somewhere through the trials. In most other circumstances, however, the carried load will prove a reason- ably good estimate of the true offered load in systems not provided with alternate paths. This is a matter of especial interest for both toll and local operation in America since principal future reliance for load measurement is ex- THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 435 pected to be placed on automatically processed TUR data, and as the TUR is a switch counting device the results will be in terms of load carried. Moreover, the quantity now obtained in many local exchanges is load carried.* Visual switch counting of line finders and selectors off- normal is widely practiced in step-by-step and panel offices; a variety of electromechanical switch counting devices is also to be found in crossbar offices. It is common to take load-carried figures as equal to load-offered when using conventional trunking tables to ascertain the proper pro- vision of trunks or switches. Fig. 7 compares the NC predictions made by a number of the available load-loss formulas when load carried is used as the entry variable. The lowest curves (1) on Fig. 7 are from the Erlang lost calls formula El (or B) with load carried L used as the offered load a. At low losses, say 0.01 or less, either L or a = L/[l — Ei(a)] can be used indiscrimi- nately as the entry in the Ei formula. If however considerably larger losses are encountered and calls are not in reality "cleared" upon meet- ing NC, it will no longer be satisfactory to substitute L for a. In this circumstance it is common to calculate a fictitious load a' to submit to the c paths such that the load carried, a'[I — Ei^dd')], equals the desired L. (This was the process used in Section 2 to obtain " % NC existing.") The curves (2) on Fig. 7 show this relation ; physically it corresponds to an initially offered load of L erlangs (or L call arrivals per average hold- ing time), whose overflow calls return again and again until successful but without disturbing the randomness of the input. Thus if the loss from this enhanced random traffic is E, then the total trials seen per holding time will be L(l + ^ + ^' -f • • •) = L/(l - E) = a', the ap- parent arrival rate of new calls, but actually of new calls plus return attempts. The random resubmission of calls may provide a reasonable descrip- tion of operation under certain circumstances, presumably when re-trials are not excessive. Kosten^ has discussed the dangers here and provided upper and lowxr limit formulas and curves for estimating the proportions of NC's to be expected when re-trials are made at any specified fixed leturn time. His lower bounds (lower bound because the change in con- gestion character caused by the returning calls is ignored) are shown by open dots on Fig. 7 for return times of 1.67 holding times. They lie above curves (2) (although only very slightly because of the relatively long return time) since they allo\\- for the fact that a call shortly returning * In fact, it is difficult to see how any estimate of offered load, other than carried load, can be obtained with useful reliability. 436 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 after meeting a busy signal will have a higher probability of again find- ing all paths busy, than would a randomly originated call. The curves (3) show the TF-formula previously developed in this sec- tion, which contemplates exponential return times on all NC attempts. The average return time here is also taken as 1 .67 holding times. These curves lie higher than Kosten's values for two reasons. First, the altered congestion due to return calls is allowed for; and second, with exponential returns nearly two-thirds of the return times are shorter than the aver- age, and of these, the shortest ones will have a relatively high probability of failure upon re-trying. If the customers were to return with exponen- tial times after waiting an average of only 0.2 holding time (e.g., 30 seconds wait for 150-second calls) the TT^-curves would rise markedly to the positions shown by (4). Curves (5) and (6) give the proportions of time that all paths are busy (equation 4) under the T'F-formula assumptions corresponding to NC curves (3) and (4) respectively; their upward displacement from the random return curves (2) reflects the disturbance to the group congestion produced by the non-random return of the delayed calls. (The limiting position for these curves is, of course, given by Erlang's E2 (or C) delay formula.) As would be expected, curve (6) is above (5) since the former contemplates exponential returns with average of 0.2 holding time, as against 1.67 for curve (5). Neither the (5)-curves nor the open dots of constant 30-second return times show a marked increase over curves (2). This appears to explain why the relationship of load carried versus "NC existing" (as charted in Figs. 3 and 4) was found so insensitive to vari- able operating procedures in handling subsequent attempts in toll ring- down operation, and again, why it did not appreciably change under operator dialing. Finally, through the two fields of curves on Fig. 7 is indicated the Poisson summation P{c, L) with load carried L used as the entering variable. The fact that these values approach closely the (2) and (3) sets of curves over a considerable range of NC's should reassure those who have been concerned that the Poisson engineering tables were not useful for losses larger than a few per cent.* 4. SERVICE REQUIREMENTS FOR DIRECT DISTANCE DIALING BY CUSTOMERS As shown by the TF-curves (3) on Fig. 7, the attempt failures by cus- tomers resulting from their tendency to re-try shortly following an NC * Reference may be made also to a throwdown by C. Clos (Ref. 3) using the return times of Fig. 6; his "% NC" results agreed closely with tlie Poisson pre- dictions. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 437 would be expected to exceed slightly the values for completely random re-trials. These particular curves are based on a re-trial interval of 1.67 times the average circuit-use time. Such moderation on the part of the customer is probably attainable through instructional literature and other means if the customer believes the "NC" or "busy" to be caused by the called party's actually using his telephone (the usual case in local practice). It would be considerably more difficult, however, to dissuade the customer from re-trying at a more rapid rate if the circuit NC's should generally approach or exceed actual called-party busies, a con- dition of which he would sooner or later become aware. His attempts might then be more nearly described by the (4) curves on Fig. 7 cor- responding to an average exponential return of only 0.2 holding time — or e\en higher. Such a result would not only displease the user, but also result in the requirement of increased switching control equipment to handle many more wasted attempts. If subscribers are to be given satisfactory direct dialing access to the iiitertoll trunk network, it appears then that the probability of finding XC even in the busy hours must be kept to a low figure. The following engineering objective has tentatively been selected: The calls offered to the ^'final" group of trunks in an alternate route system should receive no more than 3 per cent NC(P.03) during the network busy season busy hour. (If there are no alternate routes, the direct group is the "final" route.) Since in the nationwide plan there will be a final route between each of some 2,600 toll centers and its next higher center, and the majority of calls offered to high usage trunks will be carried without trying their final route (or routes), the over-all point-to-point service, while not easy to estimate, will apparently be quite satisfactory for cus- tomer dialing. 5. ECONOMICS OF TOLL ALTERNATE ROUTING In a general study of the economics of a nationwide toll switching plan, made some years ago by engineers of the American Telephone and Tele- graph Company, it was concluded that a toll line plant sufficient to give ihe then average level of service (about T-40) with ordinary single-route procedures could, if operated on a multi-alternate route basis, give the desired P.03 service on final routes with little, if any, increase in toll line investment.* On the other hand to attain a similar P.03 grade of service by liberalizing a typical intertoll group of 10 trunks working presently * This, of course, does not reflect the added costs of the No. 4 switching equip- I nient. 438 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 at a T-40 grade of service and an occupancy of 0.81 would recjuire an increase of 43 per cent (to 14.3 trunks), with a corresponding decrease in occupancy to 0.57. The possible savings in toll lines with alternate routing are therefore considerable in a system which must pro\'ide a service level satisfactory for customer dialing. In order to take fullest advantage of the economies of alternate rout- ing, present plans call for five classes of toll offices. There will be a large number of so-called End Offices, a smaller number of Toll Centers, and progressively fewer Primary Centers (about 150), Sectional Centers (about 40) and Regional Centers (9), one of which will be the National Center, to be used as the "home" switching point of the other eight Regional Centers.* Primary and higher centers will be arranged to per- form automatic alternate routing and are called Control Switching Points (CSP's). Each class of office will "home" on a higher class of office (not necessarily the next higher one) ; the toll paths between them are called "final routes." As described in Section 4, these final routes will be provided to give low delays, so that between each principal toll point and ever}' other one there will be available a succession of approximatelj' P.03 engineered trunk groups. Thus if the more direct and heavily loaded interconnecting paths commonly provided are busj- there will still be a good chance of making immediate connection over final routes. Fig. 8 illustrates the manner in which automatic alternate routing will operate in comparison with present-day operator routing. On a call from Syracuse, X. Y., to Miami, Florida, (a distance of some 1,250 miles), under present-day operation, the Syracuse operator signals Albany, and requests a trunk to Miami. With T-schedule operation the Syracuse- Miami traffic might be expected to encounter as much as 25 per cent NC during the busy hour, and approximately 4 per cent NC for the whole day, producing perhaps a two-minute over-all speed of serA-ice in the busy season. With the proposed automatic alternate routing plan, all points on the chart will have automatic switching systems. f The customer (or the operator until customer dialing arrangements are completed) will dial a ten-digit code (three-digit area code 305 for Florida plus the listed Miami seven-digit telephone number) into the Jiiachine at Syracuse. The various routes which then might conceivably be tried automatically * Sec the hihlio^rajjliy ( i);irticulMily Pilliod and Truitt) for details of tlie general trunkinji plan. t The notation uscmI on the diagram of Fig. 8 is: Opon firclo — Primary Center (Syracuse, Miamij; Triangle — Sectional Center (All)an\-, Jacksonville); Sqviare — Regional Center (White Plains, Atlanta, St. Louis; St. Louis is also the Na- tional Center). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 439 PRESENT OPERATOR ROUTING '^^ AUTOMATIC ALTERNATE ROUTING white Plains N. Y.) Miami Miami Fig. 8 — Present and proposed methods of handling a call from Syracuse, N. Y., to Miami, Florida. are shown on the diagram numbered in the order of trial; in this par- ticular layout shown, a maximum of eleven circuit groups could be tested for an idle path if each high usage group should be found NC. Dotted lines show the high usage roiites, which if found busy will overflow to the final groups represented by solid lines. The switching ecjuipment at each point upon finding an idle circuit passes on the required digits to the next machine. While the routing possibilities shown are factual, only in rare instances would a call be completed over the final route via St. Louis. Even in the busy season busy hour just a small portion of the calls would be expected to be switched as many as three times. And only a fraction of one per cent of all calls in the busy hour should encounter NC. As a result the service will be fast. When calls are handled by a toll operator, the cus- 440 THE BELL SYSTEM TECHNICAL JOURNAL; MARCH 1956 tomer will not ordinarily need to hang up when NC is obtained. When he himself dials, a second trial after a short wait following NC should have a high probability of success. Not many situations will be as complex as shown in Fig. 8; commonly several of the links between centers will be missing, the particular ones retained having been chosen from suitable economic studies. A large number of switching arrangements Avill be no more involved than the illustrative one shown in Fig. 9(a), centering on the Toll Center of Bloomsburg, Pennsylvania. The dashed lines indicate high usage groups from Bloomsburg to surrounding toll centers; since Bloomsburg "homes" on Scranton this is a final route as denoted by the solid line. As an exam- ple of the operation, consider a call at Bloomsburg destined for Williams- port. Upon finding all direct trunks busy, a second trial is made via Harrisburg; and should no paths in the Harrisburg group be available, a third and final trial is made through the Scranton group. In considering the traffic flow of a network such as illustrated at Bloomsburg it is convenient to employ the conventional form of a two- stage graded multiple having "legs" of varying sizes and traffic loads individual to each, as shown in Fig. 9(b). Here only the circuits im- mediately outgoing from the toll center are shown; the parcels of traffic (a) GEOGRAPHICAL LAYOUT WILLIAMSPORT I SCRANTON BLOOMSBURG HARRISBURG PA. (b) GRADED MULTIPLE SCHEMATIC FRACKVILLE HAZLETON WILKES- BARRE PHILADELPHIA FINAL GROUP TO SCRANTON H.U. GROUP TO HARRISBURG .1 M t I NO. TRUNKS IN H.U. GROUPS I [T] [jF] [^ [A] [T] [28 1 rsl m LOAD TO AND FROM ^^^ .^. ^^ ^ DISTANT OFFICE (CCS) "^^^ '^' ^^ ^'^^ ^^' '^0 '^3 836 228 154 DISTANT OFFICE SCRN HBG PTVL SHKN SNBY WMPT FKVL HZN WKSB PHLA Fig. 9 liiirg, Pa. Aulonialic ;ilU'riiaie routing for direct distance dialing at Blooms- THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 441 calculated for each further connecting route will be recorded as part of the offered load for consideration when the next higher switching center is engineered. It is implicitly assumed that a call which has selected one of the alternate route paths will be successful in finding the necessary paths available from the distant switching point onward. This is not quite true but is believed generally to be close enough for engineering piu'poses, and permits ignoring the return attempt problem. 6. NEW PROBLEMS IN THE ENGINEERING AND ADMINISTRATION OF INTER- TOLL GROUPS RESULTING FROM ALTERNATE ROUTING With the greatly increased teamwork among groups of intertoll trunks which supply overflow calls to an alternate route, an unexpected increase or flurry in the offered load to any one can adversely affect the service to all. The high efficiency of the alternate route networks also reduces their overload carrying ability. Conversely, the influence of an underprovision of paths in the final alternate route may be felt by many groups which overflow to it. With non-alternate route arrangements only the single groups having these flurries would be affected. Administratively, an alternate route trunk layout may well prove easier to monitor day by day than a large number of separate and in- dependent intertoll groups, since a close check on the service given on the final routes only may be sufficient to insure that all customers are being served satisfactorily. When rearrangements are indicated, how- SIMPLE PROGRESSIVE GRADED MULTIPLE GRADED MULTIPLE (a) (b) t t t t t t tt t t tl ILLUSTRATIVE INTERLOCAL AND INTERTOLL ALTERNATE ROUTE TRUNKING ARRANGEMENT; (c) (d) t t t t t = ,-"" ^ tttl It ttl 1 t Fig. 10 — Graded multi])los .•nid altornaic route trunking nrrangeinoiits. I 442 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ever, the determination of the proper place to take action, and the de sirable extent, may sometimes be difficult to determine. Suitable traffic measuring devices must be provided with these latter problems in mind For engineering purposes, it will be highly desirable: (1) To be able to estimate the load-service relationships with any specified loads offered to a particular intertoll alternate routing network; and (2) To know the day-to-day busy hour variations in the various groups' offered loads during the busy season, so that the general grade of service given to customers can be estimated. The balance of this paper will review the studies which have been made in the Bell System toward a practicable method for predicting the grade of service given in an alternate route network under any given loads. Analyses of the day-to-day load variations and their effects on customer dialing service are currently being made, and will be reported upon later. ?; 7. LOAD-SERVICE RELATIONSHIPS IN ALTERNATE ROUTE SYSTEMS In their simplest form, alternate route systems appear as symmetrical graded multiples, as shown in Fig. 10(a) and 10(b). Patterns such as these have long been used in local automatic systems to partially over- come the trunking efficiency limitations imposed by limited access switches. The traffic capacity of these arrangements has been the sub- ject of much study by theory and "throwdowns" (simulated traffic studies) both in the United States and abroad. Field trials have sub- stantiated the essential accuracy of the trunking tables which have resulted. In toll alternate route systems as contemplated in America, however, there will seldom be the symmetry of pattern found in local graded multiples, nor does maximum switch size generally produce serious limitation on the access. The ''legs" or first-choice trunk groups will vary widely in size; likewise the number of such groups overflowing calls jointly to an alternate route may cover a considerable range. In all cases a given group, whether or not a link of an alternate route, will have one or more parcels of traffic for which it is the first-choice route. [See the right-hand parcel of offered traffic on Fig. 10(c).] Often this first routed traffic will Ijc the bulk of the load offered to the group, which also serves as an alternate I'oute for other traffic. The simplest of the approximate formulas developed for solving the local graded multiple problems are hopelessly unwieldy when applied to such arrangements as shown in Fig. 10(d). Likewise it is impracticable i THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 443 to solve more than a few of the infinite variety of arrangements by means of "throwdowns." However, for both engineering (planning for future trunk provisions) I and administration (current operating) of trunks in these multi-alternate routing systems, a rapid, simple, but reasonably accurate method is (required. The basis for the method which has been evolved for Bell System use will be described in the following pages. 7.1. The "Peaked" Character of Overflow Traffic The difficulty in predicting the load-service relationship in alternate route systems has lain in the non-random character of the traffic over- flowing a first set of paths to which calls may have been randomly offered. This non-randomness is a well appreciated phenomenon among traffic engineers. If adecjuate trunks are provided for accommodating the momentary traffic peaks, the time-call level diagram may appear as in Fig. 11(a), (average level of 9.5 erlangs). If however a more limited j number of trunks, say a: = 12, is provided, the peaks of Fig. 11(a) will be Ichpped, and the overflow calls will either be "lost" or they may be j handled on a subsequent set of paths y. The momentary loads seen on 2/ then appear as in Fig. 11(b). It will readily be seen that a given average i load on the y trunks will have quite different fluctuation characteristics i than if it had been found on the x trunks. There will be more occurrences of large numbers of calls, and also longer intervals when few or no calls are present. This gives rise to the expression that overflow traffic is "peaked." Peaked traffic requires more paths than does random traffic to operate at a specified grade of delayed or lost calls service. And the increase in paths required will depend upon the degree of peakedness of the traffic involved. A measure of peakedness of overflow traffic is then required which can be easily determined from a knowledge of the load offered and the number of trunks in the group immediately available. In 1923, G. W. Kendrick, then with the American Telephone and I Telegraph Company, undertook to solve the graded multiple problem ■through an application of Erlang's statistical eciuilibrium method. His i principal contribution (in an unpublished memorandum) was to set up I the equations for describing the existence of calls on a full access group \oi X -{- y paths, arranged so that arriving calls always seek service first iu the .T-group, and then in the ^/-group when the x are all busy. Let f{m, n) be the probability that at a random instant m calls exist j on the x paths and n calls on the y paths, when an average Poisson load 444 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of a erlangs is submitted to the x -\- y paths. The general state equation for all possible call arrangements, is (a* + m + n)f{m, n) = (w + l)/(m + 1, n) + (n + l)/(m, w + 1) + ajim — 1, n) + aj{x, n — 1)% (7) in which the term marked {%) is to be included only when m = x, and * indicates that the a in this term is to be omitted when in -\- n = x -{- y. m and n may take values only in the intervals, -^ m ^ x;Q -^ n -^ y. As written, the equation represents the "lost calls cleared" situation. (a) RANDOM TRAFFIC 10 00 AM < I if) Q. a. 2 to 10 00 A M 10 30 TIME OF DAY (b) PEAKED OVERFLOW TRAFFIC PI -^ 10 30 TIME OF DAY Fig. 11 — Production of peakedness in overflow traffic. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 445 By choosing x -]- y large compared with the submitted load a a "lost calls held" situation or infinite-overflow-trunks result can be approached as closely as desired. Kendrick suggested solving the series of simultaneous equations (7) by determinants, and also by a method of continued fractions. However little of this numerical work was actually undertaken until several years later. Early in 1935 Miss E. V. Wyckoff of Bell Telephone Laboratories be- came interested in the solution of the (x -\- 1)(^/ + 1) lost calls cleared simultaneous equations leading to all terms in the /(m, n) distribution. She devised an order of substituting one equation in the next which pro- vided an entirely practical and relatively rapid means for the numerical solution of almost any set of these equations. By this method a con- siderable number of /(m, n) distributions on x, y type multiples with varying load levels were calculated. From the complete m, n matrix of probabilities, one easily obtains the distribution 9m{n) of overflow calls when exactly m are present on the lower group of x trunks; or by summing on m, the d{n) distribution with- out regard to m, is realized. A number of other procedures for obtaining the/(m, n) values have been proposed. All involve lengthy computations, very tedious for solution by desk calculating machines, and most do not have the ready checks of the WyckofT-method available at regular points through the calculations. In 1937 Kosten^ gave the following expression for /(m, n) : /(», n) = (- l)V.fe) i (i) M^- "f^'l., (8) i=0 (Pi^l{x)ipi(x) where (po{x) = x^—a a e xl ; and for i > 0, ;=o \ J / (.^• - J)i These equations, too, are laborious to calculate if the load and num- 1 K^rs of trunks are not small. It would, of course, be possible to program a modern automatic computer to do this work with considerable rapidity. The corresponding application of the statistical equilibrium equations to the graded multiple problem was visualized by Kendrick who, how- ever, went only so far as to write out the equation for the three-trunk 446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 case consisting of two subgroups of one trunk each and one common overflow trunk. Instead of solving the enormously elaborate system of equations de- scribing all the calls which could simultaneously be present in a large multiple, several ingenious methods of convoluting the X 6(n) = Z/(w, n) overflow distributions from the individual legs of a graded multiple have been devised. For example, for the multiple of Fig. 10(a), the probability of loss Pi as seen by a call entering subgroup number i, is approximately, Pi = 2 £ e.Ar)-rl^{z -r) +J: d.Ar) (9) r=0 z=y T—y in which \l/{z — r) is the probability of exactly z — r overflow calls being present, or wanting to be present, on the alternate route from all the subgroups except the zth, and with no regard for the numbers of calls present in these subgroups. The ^x,i(^) = jiixi , r) term, of course, con- templates all paths in the particular originating call's subgroup being occupied, forcing the new call arriving in subgroup i to advance to the alternate route. This corresponds to the method of solving graded mul- tiples developed by E. C. Molina^ but has the advantage of overcoming the artificial "no holes in the multiple" assumption which he made. Similar calculating procedures have been suggested by Kosten.* These computational methods doubtless yield useful estimates of the resulting service, and for the limited numbers of multiple arrangements which might occur in within-office switching trains (particularly ones of a sym- metrical variety) such procedures might be practicable. But it would be far too laborious to obtain the individual overflow distributions Q{n), and then convolute them for the large variety of loads and multiple arrangements expected to be met in toll alternate routing. 7.2. Approximate Description of the Character of Overflow Traffic It was natural that various approximate procedures should be tried in the attempt to obtain solutions to the general loss formula sufficiently accurate for engineering and study purposes. The most ol^vious of these is to calculate the lower moments or semi-invariants of the loads over- flowing th(; sul)groups, and from them construct approximate fitting * Kosten gives the above approximation (9), which he calls Wb^, Jis an upper limit to the blocking. He also gives a lower limit , Wr, in which z = // throughout (References 4, 5). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 447 I distributions for 6{n) mid dx(;n). Since each such overflow is independent I of the others, they may be combined additively (or convokited), to ob- [tain the corresponchng total distribution of calls appearing before the , I alternate route (or common group) . It may further be possible to obtain I [an approximate fitting distribution to the sum-distribution of the over- flow calls. The ordinary moments about the point of the subgroup overflow distribution, when m of the x paths are busy, are found by V ta'im) = 2 njim, n) (10) When an infinite number of |/-paths is assumed, the resulting expres- sions for the mean and variance are found to be:* Number of x-paths busy unspecified :'\ Mean = a = a-Ei,^{a) (11) Variance = v = a[l — a -{- a{x -\- I -\- a — a)'^] (12) All x-paths occupiedi Mean = a^^ = a[x - a + 1 -\- aEiMf^ (13) Variance = v^ = ax[l — ax + 2a(x + 2 + a^ — a)~^] (14) Equations (11) and (12) have been calculated for considerable ranges 1 of offered load a and paths x. Figs. 12 and 13 are graphs of these results. i For example when a load of 4 erlangs is submitted to 5 paths, the aver- I age overflow load is seen to be a = 0.80 erlang, the same value, of I course, as determined through a direct application of the Erlang Ei formula. During the time that all x paths are busy, however, the over- flow load wdll tend to exceed this general level as indicated by the value of ax = 1.41 erlangs calculated from (13). Similarly the variance of the overflow load will tend to increase when the x-paths are fully occupied, * The derivation of these equations is given in Appendix I. t The skewness factor may also be of interest : ilz l^i: 3/2 ^" + "-"^"' +a^ (15) x+1 +a- a \x + 2\{x-a)'^^-2{x-a) + x + 2 + {x^-2-a)a + 3(1 -a) I + a(l - a)(l - 2a) o K:i' \ . . . t i > . wm Mm ^ ' \'' '^ 'mV \ I ■ . \ m \ ^. \ \ q o 6 |r ly\v\\ \ . . \ • ^ \ \ \ ■ ■■ \, r- '\ iD o '^ 0) '^ * \ « , \ \ \ \ o \ F?^ \ \ X, a v^ X V^ 'S ■f 'x^ ^^ ro "^ ■ ■■■^ ■^ ."-^.^ "■\ ^^N "^ z < _] a: LU q: LU a ■ < i - o : < ) ir ■ > ■; < . lO <o o o r» lO 'f ro (M in ^f 6 6 6 6 o 6 o 6 o o o o o o o o o 6 Re) ''''3'e=1 s^Nviaa Ni'sHivd x ONissvd avon 39Vd3A\f'=» 448 o in 6 \ \ \ \ , o 'N CO IHH t t W It 1 \' , \ V '\ , ■\^ \ ■A' • ^ \ \ \ \ \, (M ' II \ \ \ \ ^,^ \ o T3 \ « Lh O ^ n c \, o3 If) 1^ o '^i "X z < _1 O o a. T— 1 ■ <^ 2 r-< ■~ +3 Q OJ cr o \ 00 UJ LL IL o O X < O ^ X o _] q3 ID OJ IIJ 0) < LU > o o ■■,,. II <o bO a> > < (M 1 t\J T— 1 X o bJQ S ^x. X ■ v>r.m,^Mt«.f,.i.,sxrrfri o o o o ~ p 6 6 6 d d S5NVla3 Nl 'SHlVd X ONISSVd QVOl 30Va3AV = » 449 SHlVd X ONISSVd avOI dO 30NVIHVA = A 450 o in 6 ■ ^^ 1:1 i:, !• > o o X o ro o \ X \ \ \ o \- \ ' \ \ s V ■■V. V x N \ \ \ \ X \ \ . \ ^ V • X \ CM ro in o z < _l a. oi o n z Q lU cc ico li- ifvj o i a < o _i UJ (M ^ CK > < o ■n \ \ ', ' II o t-- vo n r^ SHiVd X ONISSVd avO"l dO 30NVIbVA=A 451 q r~; m ^ n - 6 6 6 6 6 o bO C O O O > o o a •I— I 1— t bi) 452 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 as shown by ?; = 1.30, and Vx = 1.95. In all cases the variances v and Vx will exceed the variance of corresponding Poisson traffic (which would have variances of a and ax respectively). 7.2.1. A Prohahility Distribution for Overflow Traffic It would be of interest to be able, given the first several descriptive parameters of any traffic load (such as the mean and variance and skew- ness factors of the overflow from a group of trunks), to construct an approximate probability distribution d{n) which would closely describe the true momentary distribution of simultaneous calls. Any proposed fitting distribution for the overflow from random traffic offered to x trunks, can, of course, be compared with . ^ X determined from (7) or (8). Suitable fitting curves should give probabilities for all possitive in- tegral values of the variable (including zero) , and have sufficient unspeci- fied constants to accommodate the parameters selected for describing the distribution. Moreover, the higher moments of a fitting distribution should not diverge too radically from those of the true distribution ; that is, the "natural shapes" of fitting and true distributions should be simi- lar. Particularly desirable would be a fitting distribution form derived with some attention to the physical circumstances causing the ebb and flow of calls in an overflow situation. The following argument and der- ivation undertake to achieve these desiderata.* A Poisson distribution of offered traffic is produced by a random arrival of calls. The assumption is made or implied that the probability of a new arrival in the next instant of time is quite independent of the number currently present in the system. When this randomness (and correspond- ing independence) are disturbed the resulting distribution will no longer be Poisson. The first important deviation from the Poisson would be expected to appear in a change from variance = mean, to variance ^ * A two-parameter function which has the ability to fit quite well a wide variety of true overflow distributions, has the form t(n) = Kin + l)''e-^(''+i) in which K is the normalizing constant. The distribution is displaced one unit from the usual discrete generalized exponential form, so that ^(0) 9^ 0. The ex- pression, however, has little rationale for being selected a priori as a suitable fitting function. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 453 mean. Corresponding changes in the higher moments would also be expected. WTiat would be the physical description of a cause system with a vari- ance smaller or larger than the Poisson? If the variance is smaller, there must be forces at work which retard the call arrival rate as the number of calls recently offered exceeds a normal, or average, figure, and which increase the arrival rate when the number recently arrived falls below the normal level. Conversely, the variance will exceed the Poisson's .should the tendencies of the forces be reversed.* This last is, in fact, a rough description of the incidence rates for calls overflowing a group of trunks. Since holding times are attached to and extend from the call arrival instants, calls are enabled to project their influence into the future; that is, the presence of a considerable number of calls in a system at any in- stant reflects their having arrived in recent earlier time, and now can be used to modify the current rate of call arrival. Let the probability of a call originating in a short interval of time dt be Po.n = [a + (n — a)co(n)] dt where n = number of calls present in the system at time t, a = base or average arrival rate of calls per unit time, and w(n) = an arbitrary function which regulates the modification in call origination rate as the number of calls rises above or falls below a. Correspondingly, let the probability that one of n calls will end in the short interval of time dt be which will be satisfied in the case of exponential call holding times, with mean unity. Following the usual Erlang procedure, the general statistical equilibrium equation is (16) Jin) = /(n)[(l - Po.n){l - Pe,n)\ + /(« " l)Po,n-l(l " Pe.n-l) -Vj{n+ 1)(1 - Po.„+i)P,,„+i which gives (Po,„ + P.,„)/(n) = Po,«-i/(n - 1) + Pe,n+xKn + 1) i ignoring terms of order higher than the first in dt. * The same thinking lias been used by Vaiilot^ for decreasing the call arrival I rate according to the number momentaril}^ present; and by Lundquist^ for both increasing and decreasing the arrival rate. 454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 (17) Or, [a + (n — a)w(?i) + ??.]/(n) = [a + (n -a- l)co(n - l)]/(/^ - 1) + (n + l)/(7i + 1) The choice of aj(n) will determine the solution of (17). Most simply, co(n) = k, making the variation from the average call arrival rate directly- proportional to the deviation in numbers of calls present from their average number. In this case, the solution for an unlimited trunk group becomes, with a' = a{l — k), a (a + k) -■■ [a + {n - 1)A;] fin) = n! ^^ , , a' (a' + k) , a' (a' + k)(a' + 2k) , 1 + « H ^t; H ^ TT, + (18) 2! ' 3! which may also be written after setting a" = a'/k = a(l — k)/k, as a'ia' + 1) • • • [a" + (n - 1)]A;" fin) = n! (19); (1 - k)- The generating function (g.f.) of (19) is Z/(n)r = (1 - kT)-"" n=0 (1 - k)--" which is recognized as that for the negative binomial, as distinguished from the g.f., P (i + ? tX (1/g)^ for the positive binomial. The first four descriptive parameters of /(w) are: Order Moment about Mean Descriptive Parameter 1 Ml = Mean = n = a (20) 2 M2 = variance, v = a/(l — k) Std Devn, <r = [a/(l - fc)]'/2 (21) 3 f^a a(l + k) (1 - fc)^ Skewness, \/sT — — , , (22) 4 M4 3a2(l - A0 + a(fc2+4A; + l) M4 A;2 + 4fc + 1 Kurtosis, /3., = - = 3 + —7^ ry- o-* a(l — k) (23) (1 - fc)3 I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 455 Since only two constants, a and k, need specification in (18) or (19), the mean and variance are sufficient to fix the distribution. That is, with the mean /7 and variance v known, a = ,7 or a' = n(l - k) = if/v, or a" = n(l - k)/k (24) A: = 1 - a/y = 1 - n/v. (25) The probability density distribution f(n) is readily calculated from (19); the cumulative distribution G(^n) also may be found through use of the Incomplete Beta Function tables since G(^n) = hi7i - l,a") (26) = h(n - l,a(l - k)/k) The goodness with which the negative binomial of (19) fits actual dis- tributions of overflow calls requires some investigation. Perhaps a more elaborate expression for co(n) than a constant k in (17) is required. Three comparisons appear possible: (1), comparison with a variety of 0«(n) distributions with exactly m calls on the x trunks, or d{n) with m unspeci- fied, (obtained by solving the statistical equilibrium equations (7) for a divided group) ; (2), comparison with simulation or "throwdown" results; and (3), comparison with call distributions seen on actual trunk groups. These are most easily performed in the order listed.* Co7nparison of Negative Binomial with True Overflow Distributions Figs. 14 to 17 show various comparisons of the negative binomial dis- tribution with true overflow distributions. Fig. 14 gives in cumulative form the cases of 5 erlangs offered to 1, 2, 5, and 10 trunks. The true j = n distributions (shown as solid lines) are obtained by solving the difference equations (7) in the manner described in Section 7.1. The negative bi- nomial distributions (shown dashed) are chosen to have the same mean and variance as the several F{^n) cases fitted. The dots shown on * Comparison could also be made after equating means and variances respec- tively, between the higher moments of the overflow traffic beyond x trunks and the corresponding negative binomial moments: e.g., the skewness given by (15) can be compared with the negative binomial skewness of (22). The difficulty here is that one is unable to judge whether the disparity between the two distribution functions as described by differences in their higher parameters is significant or not for traffic engineering purposes. 456 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 the figure are for random (Poisson) traffic having the same mean values as the /'' distributions. The negative binomial provides excellent fits down to cumulated probabilities of 0.01, with a tendency thereafter to give somewhat larger values than the true ones. The Poisson agreement is good only for the overflow from a single trunk, as might have been anticipated, the divergence rapidly increasing thereafter. Fig. 15 corresponds with the cases of Fig. 14 except that the true over- flow Fxi^n) distributions for the conditional situation of all .r-paths busy, are fitted. Again the negative binomial is seen to give a good agree- ment down to 0.01 probability, with somewhat too-high estimates for larger values of the simultaneous overflow calls n. Fig. 16 shows additional comparisons of overflow and negative bi- nomial distributions. As before, the agreement is quite satisfactory to 0.01 probability, the negative binomial thereafter tending to give some- what high values. On Fig. 17 are compared the individual 6(n) density distributions for several cases. The agreement of the negative binomial with the true distribution is seen to be uniformly good. The dots indicate the random (Poisson) individual term distribution corresponding to the a = 9.6 case- 1.0 "T*^ ;J-^ — TRUE DISTRIBUTION \ ^^^^^\- _ NEGATIVE BINOMIAL \ <^ • \ FITTING DISTRIBUTION CORRESPONDING RANDOM TRAFFIC 0.1 -\ \ > v \ _ \ • \ •\ \ n) \ \ ^ \ » \\ \ \ • \ ^ V \ 0.01 - \ V5 • v\ \s:=io \\ \\ \ . ^ V \> V • \ • \ ^^ \ n^ 0.001 _J M \ i 1 \ l> \V 1 1 t 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n = NUMBER OF SIMULTANEOUS CALLS Fig. 14 — Probability distributions of overflow traffic with 5 erlangs offered to 1, 2, 5, and 10 trunks, fitted by negative binomial. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 457 the agreement, of course, is poor since the non-randomness of the over- flow here is marked, having an average of 1.88 and a variance of 3.84. Comparison of Negative Binomial with Overflow Distributions Observed hi/ llirowdoivns and on Actual Trunk Groups Fig. 18 shows a comparison of the negative binomial with the over- How distributions from four direct groups as seen in throwdown studies, 'ilie agreement over the range of group sizes from one to fifteen trunks is seen to be excellent. The assumption of randomness (Poisson) as shown by the dot values is clearly unsatisfactory for overflows beyond more than two or three trunks. A number of switch counts made on the final group of an operating toll alternate routing system at Newark, New Jersey, during periods when few calls were lost, have also shown good agreement with the neg- ative binomial distribution. 7.2.2. A Probability Distribution for Combined Overflow Traffic Loads It has been shown in Section 7.2.1 that, at least for load ranges of wide interest, the negative binomial with but two parameters, chosen to agree Fx(§n) 0.01 0.001 TION OMIAL BUTION I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n= NUMBER OF SIMULTANEOUS CALLS Fig. 15 — Probability distributions of overflow traffic with 5 erlangs offered to 1, 2, 5, and 10 trunivs, when all trunks are busy; fitted by negative binomial. 458 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 with mean and variance, gives a satisfactory jfit to the distribution of traffic overflowing a group of trunks. It is now possible, of course, to convohite the various overflows from any number of groups of varying sizes, to obtain a combined overflow distribution. This procedure, how- ever, would be very clumsy and laborious since at each switching point in the toll alternate route system an entirely difl"erent layout of loads and high usage groups would require solution; it would be unfeasible for practical working. We return again to the method of moments. Since the overflows of the several high usage groups will, in general, be independent of one another, the iih semi-invariants Xi of the individual overflows can be combined to give the corresponding semi-invariants A, of their total, Ai — iXi + 2X1 + (27) Or, in terms of the overflow means and variances, the corresponding parameters of the combined loads are Average = A' = ai -{- az + ■ ■ ■ (28) Variance = V = vi + V2 + • • - (29) TRUE DISTRIBUTION NEGATIVE BINOMIAL FITTING DISTRIBUTION 0.001 2 3 4 5 6 7 8 9 10 II 12 13 14 15 n = NUMBER OF SIMULTANEOUS CALLS Fig. 16 — Probability distributions of overflow traffic: 3 erlangs offered to 2 trunks, and 9.6 erlangs offered to 10 trunks. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 459 With the mean and variance of the combined overflows now deter- mined, the negative binomial can again be employed to give an approxi- mate description of the distribution of the simultaneous calls (p{z) offered to the common, or alternate, group. The acceptability of this procedure can be tested in various ways. One way is to examine whether the convolution of several negative binomials (representing overflows from individual groups) is sufficiently well fitted by another negative binomial with appropriate mean and variance, as found above. It can easily be shown that the convolution of several negative bi- nomials all with the same over-dispersion (variance-to-mean ratio) but not necessarily the same mean, is again a negative binomial. Shown in Table I are the distribution components and their parameters of two examples in which the over-dispersion parameters are not identical. The third and fourth semi-invariants of the fitted and fitting distributions, are seen to diverge considerably, as do the Pearsonian skewness and kurtosis factors. The test of acceptability for traffic fluctuation description comes in comparing the fitted and fitting distributions which are shown on Fig. 19. Here it is seen that, despite what might appear alarming dis- 0(n) 0.01 O.OOI TRUE DISTRIBUTION NEGATIVE BINOMIAL FITTING DISTRIBUTION • RANDOM TRAFFIC, 8=1.9 a = 9.6 = 3.84 I 2 3 4 5 6 7 8 9 10 II 12 n = NUMBER OF SIMULTANEOUS CALLS Fig. 17 — Probability density distributions of overflow traffic from 10 trunks, fitted by negative binomial. 460 THE BELL SYSTEM TECHNICAL JOUENAL, MARCH 1956 parities in the higher semi-invariants, the agreement for practical traffic purposes is very good indeed. Numerous throwdown checks confirm that the negative binomial em- ploying the calculated sum-overflow mean and variance has a wide range over which the fit is quite satisfactory for traffic description purposes. Fig. 20 shows three such trunking arrangements selected from a con- siderable number which have been studied by the simulation method. Approximate!}^ 5,000, 3,500, and 580 calls were run through in the three examples, respective!}' . Tlie overflow parameters obtained !)y experiment are seen to agree reasonably well with the theoretical ones from (28) and (29) when the numbers of calls processed is considered. On Fig. 21 are sliown, for the first arrangement of Fig. 20, distributions of simultaneous offered calls in each subgroup of trunks compared with the corresponding Poisson; the agreement is satisfactory as was to be expected. The sum distribution of the overflows from the eight subgroups is given at the foot of the figure. The superposed Poisson, of course, is a poor fit; the negative binomial, on the other hand, appears quite accept- able as a fitting curve. 1.0 0.8 0.6 P 2n 1 TRUNK- a = \.22 3 TRUNKS- a = 2.24 0.4 - 0.2 ■ 1.0 0.8 - 0.6 0.4 0.2 234501 234 n=NUMBER OF SIMULTANEOUS CALLS THEORY OBSD V\ ( ) ( ) AVG 0.67 0.63 VAR 0.77 0.60 i • RANDOM TRAFFIC \, a = 0.67 THEORY OBSD c- ) ( — 1 u AVG 0.55 0.51 VAR 0.77 0.63 \\ • RANDOM TRAFFIC a= D.55 v^^ P^n 1.0 15 TRUNKS- a \ THEORY = 11.46 OBSD '-O .\ ( H ( ) 0.8 *\ AVG 0.81 0.80 '-'•® 'A VAR 1.88 1.42 0.6 "\\, • RANDOM TRAFFIC °-^ \l a=o.8i 0.4 0.4 0.2 0.2 • ^'^v,.^^^^ _ , n 9 TRUNKS- a = 6.21 THEORY OBSD ( -) ( ) AVG 0.52 0.46 VAR 1.00 1.48 . RANDOM TRAFFIC a = 0.52 4 68 10 024 68 n=NUMBER OF SIMULTANEOUS CALLS 10 12 Fifj;. 18 — Ovorflow (li.-<t ril)utioiis from diroct interoffice trunk groups; negative binomial theory versus thrgwclowji observations. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 461 Table I — Comparison of Parameters of a Fitting Negative Binomial to the Convolution of Three Negative Binomials Example No. 1 Example No. 2 Component Component parameters Component dist'n No. Component parameters dist'n No. Mean Variance Mean Variance 1 5 5 1 1 1 2 2 4 2 2 3 3 1 3 3 2 6 8 12 5 10 Semi-Invariants A, Skewness \/pi , and Kurtosis ^2 , of Sum Distributions Parameter E.xact Fitting Parameter Exact Fitting Ai 8 8 Ai 5 5 Ao 12 12 A2 10 10 As 32 24 As 37 30 A4 168 66 A4 239.5 130 VFi 0.770 0.577 V/3i 1.170 0.949 /32 4.167 3.458 /32 5.395 4.300 Fig. 22 shows the corresponding comparisons of the overflow loads in the other two trunk arrangements of Fig. 20. Again good agreement with the negative binomial is seen. 7.3. Equivalent Random Theory for Prediction of Amount of Traffic Over- flowing a Single Stage Alternate Route, and Its Character, with Lost Calls Cleared As discussed in Section 7.2, when random traffic is offered to a limited number of trunks x, the overflow traffic is well described (at least for traffic engineering purposes) by the two parameters, mean a and variance V. The result can readily be applied to a group divided (in one's mind) two or more times as in Fig. 23. Employing the a and v curves of Figs. 12 and 13, and the appropriate numbers of trunks a;i , Xi + 0:2 , and Xi + X2 + x^ , the pairs of descrip- tive parameters, ai , vi , ao , vo and a-s , v-a can be read at once. It is clear then that if at some point in a straight multiple a traffic with parameters ai , Vi is seen, and it is offered to .r2 paths, the overflow therefrom will have the characteristics 012 , vo . To estimate the particular values of a-y and v-i , one would first determine the values of the equivalent random 462 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 P5n P^n CONVOLUTION OF 3 NEGATIVE BINOMIAL VARIABLES WITH PARAMETERS: AVG WR 1 I 2 3 2 6 , -FITTING NEGATIVE BINOMIAL 6 8 10 12 n= NUMBER OF CALLS PRESENT I -I I l_^ 14 16 Fig. 19 — Fitting sums of negative binomial variables with a negative binomial. traffic a and trunks .Ti which would have produced ai and Vi . Then pro- ceeding in the forward direction, using a and Xi + X2 , one consults the a and v charts to find txi and Vz . Thus, within the limitations of straight group traffic flow, the character (mean and variance) of any overflow load from x trunks can be predicted if the character (mean and variance) of the load submitted to them is known. Curves could be constructed in the manner just described by which the overflow's a' and v' are estimated from a load, a and v, offered to x trunks. An illustrative fragment of such curves is shown in Appendix II, with an example of their application in the calculation of a straight trunk group loss by considering the successive overflows from each trunk as the offered loads to the next. Enough, perhaps, has been shown in Section 7.2 of the generally ex- cellent descriptions of a variety of non-random traffic loads obtainable by the use of only the two parameters a and v, to make one strongly suspect that most of the fluctuation information needed for traffic engi- neering purposes is contained in those two values. If this is, in fact, the case, we should then be able to predict the overflow a', v' from x trunks THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 463 \\ith an offered load a, v which has arisen in any manner of overflow from earlier high usage groups, as illustrated in Fig. 24. This is found to be the case, as will be illustrated in several studies de- scribed in the balance of this section. In the determination of the charac- teristics of the overflow traffic a', v' in the cases of non-full-access groups, such as Figs. 24(b) and 24(c), the equivalent straight group is visualized [Fig. 24(a)], and the Eciuivalent Random load A and trunks S are found.* I Using A, and *S + C, to enter the a and v curves of Figs. 12 and 13, a , and v' are readily determined. To facilitate the reading of .1 and S, Fig. 25 1 and Fig. 26 f (which latter enlarges the lower left corner of Fig. 25) have been drawn. Since, in general, a and v will not have come from a simple straight group, as in Fig. 24(a), it is not to be expected that *S, OVERFLOW THEORY OBSD AVERAGE 5.76 5.98 VARIANCE 12.37 14.89 = = _ = OST N0.1 t t 13.16 1024 f 1024 t t 10.18 9.22 t t 7.63 7.48 0.76 ERLANGS OVERFLOW AVERAGE VARIANCE THEORY 5.02 9.95 OBSD 5.06 7.90 ^^ ~ ^ — OST N0.6 t t t t t t t OFFER 10.66 3.24 2.44 11.46 9.81 9.59 1.42 ERLANGS OVERFLOW THEORY OBSD AVERAGE 2.83 2.87 VARIANCE 3.35 3.34 OST N0.14 t t t t 1 1 t OFFER 2.52 1.08 0.94 0.94 0.59 1.13 0.85 ERLANGS Fig. 20 — Comparison of joint-overflow parameters; theory versus throwdown. * A somewhat similar method, commonly identified with the British Post Office, which uses one parameter, has been employed for solving symmetrical graded multiples (Ref. 9). t Figs. 25 and 26 will be found in the envelope on the inside back cover. THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 GROUP N0.1 17 TRUNKS, a = 13.t6 GROUP NO. 2 14 TRUNKS, a: 10 15 20 5 10 15 r = NUMBER OF SIMULTANEOUS CALLS OFFERED TO THE DIRECT TRUNKS 0.15 0.10 f(r) ao5 I- z ui BC a. Ill S 1- z o o o a. a. 10 GROUP NO. 3 13 TRUNKS, a = 10.24 q: < 1- A GROUP NO. 5 A\^^ 12 TRUNKS, a= 9.22 10 0.10 11 f(r) O 0.05 //v. c y^ ^x^\_^ a 20 0.20 r GROUP NO. 7 /\ 10 TRUNKS, a = 748 0.15 /7X\ f(r) 0.10 / nk 0.05 yy \v ---^r ^^^^^C::^— -^ GROUP N0.4 ' 14 TRUNKS,; GROUP N0.6 10 TRUNKS, i 8 10 12 14 16 18 0.15 r 0.10 F(n) 0.05 DISTRIBUTION OF OVERFLOW CALLS FROM 8 DIR GROUPS OFFERED TO 1ST ALTERNATE ROUTE THEORY OBSD AVG 5.76 5.98 VAR 12.4 14.9 THROWDOWN OBSNS NEGATIVE BINOMIAL -^^^^l!!^^^^^>^ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 n = NUMBER OF SIMULTANEOUS CALLS OFFERED TO THE ALTERNATE ROUTE 17 Fig. 21 — Comparison of theoretical and throwdown dis(ril)utions of simul- taneous calls offered to direct groups and to tlieir first alternate route (OST No. 1). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 465 read from Fig. 25, will be an integer. This causes no trouble and S should be carried along fractionally to the extent of the accuracy of result de- sired. Reading *S' to one-tenth of a trunk will usually be found sufficient for traffic engineering purposes. Example 1: Suppose a simple graded multiple has three trunks in each of two subgroups, which overflow to C common trunks, where C = 1, P^n OST NO. 6 THEORY OBSD AVG 5.02 5.06 VAR 9.95 7.90 • RANDOM TRAFFIC, a = 5.0 -OBSD -NEGATIVE BINOMIAL 2 4 6 8 10 12 14 16 18 n = NUMBER OF SIMULTANEOUS CALLS P?n --OBSD OST N0.14 THEORY OBSD ( ) ( ) AVG 2.83 2.87 VAR 3.35 3,34 RANDOM TRAFFIC, a = 2.8 -NEGATIVE BINOMIAL 2 4 6 8 10 12 14 16 18 n = NUMBER OF SIMULTANEOUS CALLS Fig. 22 — Combined overflow loads off'ered to alternate-route OST trunks from lirect interoffice trunks; negative binomial theory vs throwdown observations. t«3. V, ta2,y 2y^2 f a,,i Fig. 23 — A full access group divided at several points to examine the traffic character at each point. 466 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 2 or 3. A load of a erlangs is submitted to each subgroup, a having the values 1, 2, 3, 4 or 5. What grade of service will be given? Solution: The load overflowing each subgroup, when a = 1 for example, has the characteristics a = 0.0625 and y = 0.0790. Then A' = 2a = 0.125 and V — 2v = 0.158. Reading on Fig. 26 gives the Ecjuivalent Random values oi A = 1.04 erlangs, S = 2.55 trunks. Reading on Fig. 12.1 with C + *S = 3.55 when C = 1, and A = 1.04, we find a' = 0.0350 and oi' liflx + a-^ = 0.0175. We construct Table II in which loss values pre- dicted by the Equivalent Random (ER) Theory are given in columns (3), (5) and (7). For comparison, the corresponding exact values given by Neovius* are sho\vn in columns (2), (4) and (6). (Less exact loss s (OR X) (a) ta,v (b) fa'.v ta,v (c) fa'.V |A fa, f; la, faafaa 134*35* J Fig. 24 — Various high usage trunk group arrangements producing the same total overflow a, v. figures were given previously by Conny Palm^°. The agreement is seen to be excellent for engineering needs for all values in the table. Example 2: Suppose in Fig. 24(b) the random offered loads and paths are as given in Table III; we desire the proportion of overflow and the overflow load characteristics from an alternate route of 5 trunks. Solution: The individual overflows ai , vi ; a^ , v-i ; and as , Vz are read from Figs. 12 and 13 and recorded in columns (4) and (5) of the table. The a and v columns are totalled to obtain the sum-overflow average A' and variance V . The Equivalent Random load A which, if submitted to S trunks would produce overflow A', V , is found from Fig. 26. Finally, with A submitted io S -\- C trunks the characteristics a' and y', of the load overflowing the C trunks are found. The numerical values obtained * Artificial Traffic Trials Using Digital Computers, a paper presented by G. Neovius at the First International Congress on the Application of the Theory of Probability on Telephone Engineering and Administration, Copenhagen, June, 1955. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 467 Table II^ — Calculation of Loss in a Simple Graded Multiple g = 2, Xi = X2 = S, ai = a2 = a = 1 to 5, C = 1 to 3 T nafl Submitted to each Proportion of Each Subgroup Load which Overflows = a'/(.ai + ai) Subgroup in Erlangs a C = 1 C = 2 C = 3 True ER True ER True ER (1) 1 2 3 5 5 (2) 0.01737 0.11548 0.24566 0.35935 0.44920 (3) 0.0175 0.115 0.246 0.363 0.445 (4) 0.00396 0.05630 0.16399 0.27705 0.37336 (5) 0.0045 0.057 0.163 0.279 0.370 (6) 0.00077 0.02438 0.10212 0.20535 0.30308 (7) 0.00088 0.024 0.103 0.210 0.305 for this example are shown in the lower section of Table III. As before, of course, the "lost" calls are assumed cleared, and do not reappear in the system. Example 3: A load of 18 erlangs is offered through four groups of 10-point selector switches to twenty- two trunks which have been desig- nated as "high usage" paths in an alternate route plan. Which of the trunk arrangements shown in Fig. 27 is to be preferred, and to what extent? Solution: By successive applications of the Equivalent Random method the overflow percentages for each of the three trunk arrange- ments are determined. The results are shown in column 2 of Table IV. The difference in percentage overflow between the three trunk plans is small; however, plan 2 is slightly superior followed by plans 3 and 1 in Table III — Calculation of Overflows from a Simple Alternate Route Trunk Arrangement Subgroup Number Offered Load in Erlangs a Number of Trunks X Overflow Loads a V 1 2 3 3.5 5.7 6.0 15.2 3 6 9 1.41 1.39 0.45 3.25 1.98 2.40 0.85 5.23 Description of load offered to alternate route: A' = 3.25, V = 5.23. ]'"quivalent straight multiple: S = 5.8 trunks, A = 8.00 erlangs (from Fig. 26). Overflow from C = 5 alternate route trunks (enter Figs. 12 and 13 with A = 8.0 and S + C = 10.8: a' = 0.72, v' = 1.48. Proportion of load to commons which overflows = 0.72/3.25 = 0.22. Proportion of offered load which overflows = 0.72/15.2 = 0.0475. 468 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 PROPORTION OVERFLOWING N0.1 E.R THEORY NEOVIUS THROWDOWNS f — ^- • • • • —m- • . • . A = 18 <^ l^ — •- • • • » [ 1 BESK PUNCHED 1 CARDS ■ ■ » -*- 0.123 0.118 0.114 NO. 2 A = 18 < fr: : : 1 1 1 l~: : : n I ■ 1 -^0.113 0.110 0.110 N0.3 f"*' ■ 1 1 n 1 1 -»-0.118 0.113 0.111 l::::imii ' Fig. 27 — Comparison of losses on three graded arrangements of 22 trunks. that order. The results of extensive simulations made by Neovius on the three trunk plans are available for comparison.* The values so obtained are seen to be very close to the ER theoretical ones ; moreover the same order of preference among the three plans is indicated and with closely similar loss differentials between them. 7.3.1. Throwdown Comparisons with Equivalent Random Theonj on Simple Alternate Routing Arrangements with Lost Calls Cleared Results of manuallj' run throwdowns on a considerable number of non-symmetrical single-stage alternate route arrangements are available. Some of these were shown in Fig. 20; they represent part of a projected multi-alternate route layout (to be described later) for outgoing calls from the local No. 1 crossbar Murray Hill-6 office in New York to all other offices in the metropolitan area. The paths hunted over initially are called direct trunks; they overflow calls to Office Selector Tandem (GST) groups, numbered from 1 to 17, which are located in widely dispersed central office buildings in the Greater New York area. * Loc. cit. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 469 Table IV — Loss Comparison of Graded Arrangements Estimates of Percentage of Load Overflowing Plan Number ER Theory Neovius Throwdowns BESK Computer (262144 calls) Punched Cards (10,000 calls) (1) 1 2 3 (2) 12.3 11.3 11.8 (3) 11.81 10.98 11.25 (4) 11.4 11.0 11.1 Table V — Comparison of Theory and Throwdowns for the Parameters of Loads Overflowing the Common Trunks in Single-Stage Graded Multiples OST (Alternate) Route Group No. of Groups of Total No. of Trunks Total Load Offered to Direct Trunks Total Overflow Load from OST Group No. of trunks Direct Trunks in Direct Groups Erlangs Approximate No. of Calls Theory Throwdown no. (in 2.7 hours) a' v' a' v' 1 6 8 91 68.91 4950 2.00 5.50 2.36 6.52 2 3 3 45 37.49 2690 2.10 5.60 2.05 6.36 3 6 6 80 60.62 4355 1.50 4.00 1.30 5.67 4 3 6 52 38.49 2765 2.30 5.20 2.08 6.43 5 3 3 17 12.51 900 0.45 0.83 0.49 1.02 6 4 7 64 48.62 3490 2.50 5.90 2.36 4.88 7 8 12 78 57.42 4125 2.20 5.60 1.71 4.08 8 6 9 16 12.96 930 0.82 1.63 0.81 1.11 9 1 2 22 16.96 1220 1.30 2.60 1.02 1.73 10 5 6 10 9.52 685 0.78 1.40 1.05 2.07 11 8 13 16 16.43 1180 1.90 3.80 2.77 7.29 12 8 9 2 6.88 495 0.70 1.30 0.81 1.83 13 5 15 33 21.42 1540 1.75 3.30 1.16 2.01 14 2 7 11 8.05 580 1.46 2.20 1.63 2.14 15 9 15 8 11.97 860 1.60 3.25 1.55 4.12 16 11 22 34 27.46 1970 1.75 4.00 1.34 2.26 17 3 7 4 5.81 420 1.53 2.31 1.43 1.80 26.64 58.42 25.92 61.32 In Table V are given certain descriptive data for the 17 OST trunk arrangements showing numbers of legs of direct trunks, total direct trunks, the offered erlangs and calls, and the mean and variance of the alternate routes' overfiovvs, as obtained by the ER theory and by throwdowns.* The throwdown a' and v' values of the OST overflow * Additional details of this simulation study are given in Section 7.4. 470 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 i.a ^O 0.2 EQUIVALENT RANDOM THEORY ERLANG THEORY- 1 2 3 4 5 6 7 8 9 10 It 12 13 14 15 16 17 ALTERNATE ROUTE (OST) NUMBER Fig. 28 — Comparison of theoretical and throwdown overflows from a number of first alternate routes. were obtained by 36-second switch counts of those calls from each OST group which had come to rest on subsequent alternate routes. On Fig. 28 is shown a summary of the observed and calculated pro- portions of "lost" to "offered" traffic at each OST alternate route group. As may be seen from the figure and the last four columns of Table V, the general agreement is quite good ; the individual group variations are probably no more than to be expected in a simulation of this magnitude. An assumption of randomness (which has sometimes been argued as returning when several overflows are combined) for the load offered to the OST's gives the Erlang Ei loss curve on Fig. 28. This, as was to be expected, rather consistently understates the loss. Since "switch-counts" were made on the calls overflowing each OST, the distributions of these overflows may be compared with those esti- mated by the Negative Binomial theory having the mean and variance predicted abo\'e for the overflow. Fig. 29 shows the individual and cumu- lative probability distributions of the overflow simultaneous calls from the first two OST alternate routes. As will be seen, the agreement is quite good even though this is traffic which has been twice "non-ran- domized." Comparison of the observed and calculated overflow means and variances in Table V indicates that similar agreement between observed and theoretical fitting distributions for most of the other OST's would be found. 7.3.2. Comparison of Equivalent Random Theory with Field Results on Simple Alternate Routing Arrangements _ Data were made available to the author from certain measurements made in 1941 by his colleague C. Clos on the automatic alternate routing trunk arrangement in operation in the Murray Hill-2 central office in New York. Mr. Clos observed for one busy hour the load carried on THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 471 several of its OST alternate rovite groups (similar to those shown in Table V for the Murray Hill-6 office, but not identical) by means of an electromechanical switch-counter having a six-second cycle. During each hour's observation, numbers of calls offered and overflowing were also recorded. Although the loads offered to the corresponding direct trunks which ()^'erflowed to the OST group under observation were not simultaneously measured, such measiu'ements had been made previously for several hours so that the relative contribution from each direct group was closely known. In this way the loads offered to each direct group which produced the total arriving before each OST group could be estimated with considerable assurance. From these direct group loads the character (mean and variance) of the traffic offered to and overflowing the OST's was predicted. The observed proportion of offered traffic which over- flowed is shown on Fig. 30 along with the Equivalent Random theory prediction. The general agreement is again seen to be fairly good al- though with some tendency for the ER theory to predict higher than observed losses in the lower loss ranges; perhaps the disparity on in- (n) 0.5 0.4 0.3 0.2 0.1 OST N0.1 THEORY OBSD AVG VAR 2.00 2.36 5.50 6.52 RANDOM TRAFFIC ^--NEGATIVE BINOMIAL -THROWDOWN OST NO. 2 THEORY OBSD AVG VAR 2.10 5.60 2.05 6.36 >RAND0M TRAFFIC THROWDOWN -NEGATIVE BINOMIAL 10 15 5 n = NUMBER OF SIMULTANEOUS CALLS 15 p^n -NEGATIVE BINOMIAL -THROWDOWN -NEGATIVE BINOMIAL THROWDOWN 10 15 5 n = NUMBER OF SIMULTANEOUS CALLS 15 Fig. 29 — Distributions of loads overflowing from first alternate (OST) groups; negative binomial theory versus throwdown observations. 472 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G dividual OST groups is within the limits one might expect for data based on single-hour observations and for which the magnitudes of the direct group offered loads required some estimation. The assumption of random traffic offered to the OST gives, as anticipated, loss predictions (Erlang £"1) consistently below those observed. More recently extensive field tests have been conducted on a working toll automatic alternate route system at Newark, New Jersey. High usage groups to seven distant large cities o\'erflowed calls to the New- ark-Pittsburgh alternate (final) route. Data describing the high usage groups and typical system busy hoiu- loads are given in Table ^T. (The loads, of course, varied considerably from day to day.) The size of the Pittsburgh route varied over the six weeks of the 1955 tests from 64 to 71 trunks. Altogether the system comprised some 255 intertoll trunks. Observations were made at the Newark end of the groups by means of a Traffic Usage Recorder — making switch counts every 100 seconds — and by peg count and o^'erflow registers. Register readings were photo- graphically recorded by half-hourly, or more frequent, intervals. To ^- <a uz ^^ zz 05 1-0 (T-l °^ 1.0 0.5 0.2 2 0.1 0.05 0.02 0.01 - - - Z' - ^,^!^ 1^ f^-^^ U^ — - "^ Jft NON-RANDOM (ER) //'' THEORY "^xX /' y^ Jrf -OBSERVED X ^-i^sM ' _ X ^____— --sss:^*^^'^^^ / - /j^ ' ^^ ^'^-RANDOM THEORY ^ 1 yJ 1 / J ^ • • 11 (\j — ^ ro tn n — — ^ d d d TANDEM z Z Z z z z Z Z Z OFFICE <£> r^ t: ro CI (^ m O) ro Q Q n LU UJ UJ LU LU CD LU (D 03 NO. TRUNKS 13 12 8 7 3 8 3 4 3 OFFERED JavG 7.55 7.19 5.22 3.81 2.06 7.79 2.36 4.09 2.4 LOAD |vAR 13.58 15.66 6.59 7.30 2.51 18.54 2.77 4.59 5.90 Fig. 30 — -Observed tandem ovciflow.s in nlicriKilc llill-2 (New York) 1940-1941. loulc study at Murray THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 473 Table VI — High Usage Groups and Typical System Busy Hour Loads High Usage Group, Newark to: Length of Direct Route (Air Miles) Nominal Size of Group (Number of Trunks) Typical Offered Load (erlangs) Baltimore 170 560 395 1375 470 1100 1170 18 42 27 33 37 26 5 19 Cincinnati Cleveland Dallas Detroit Kansas City New Orleans 43 26 34 36 23 4 compare theory with the observed overflow from the final route, esti- mates of the offered load A' and its ^-ariance V are required. In the present case, the total load offered to the final route in each hour was estimated as A' = Average of Offered Load Peg Count of Calls Offered to Pittsburgh Group (Peg Count of Offered Calls) — (Peg Count of Overflow Calls) X Average Load Carried by Pittsburgh Group The variance V of the total load offered to the final route was estimated for each hour as V' = Variance of Offered Load 7 7 = A' — 2 «i + 2 Vi i=l where «» and Vi are, respectively, the average and variance of the load overflowing from the tth high usage group. (The expression. A' — 7 ^ «i , is an estimate of the average — and, therefore of the variance 1=1 — of the first-routed traffic offered directly to the final route. Thus the total variance, V, is taken as the sum of the direct and overflow com- ponents.) Using A', V and the actual number, C, of final route trunks in service, the proportion of offered calls expected to overfloAv was calcu- lated for the traffic and trunk conditions seen for 25 system busy hours from February 17 to April 1, 1955 on the Pittsburgh route. The results are displayed on Fig. 31, where certain traffic data on each hour are given in the lower part of the figure. The hours are ordered — for con- venience in plotting and viewing — by ascending proportions of calls overflowing the group; observed results are shown by the double line 474 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 0.001 3 5 7 9 11 13 15 17 19 21 23 25 NO. P'BGH TRKS 71 70 65 71 65 71 65 69 64 64 70 65 64 71 68 65 64 65 64 70 65 65 65 65 65 HOURS BY AMT. OF OBS'D loss EST'd LOAD fAVG. 50 54 55 56 55 63 55 58 54 54 68 60 63 74 76 74 76 83 91 102 109105 101 115124 OFFERED War. 82 95 85 89 98 101 84 98 97 89110 10588125 121140114 141 175182 170 176 179 199197 Fig. 31 — Final route (Newark-Pittsburgh) overflows in 1955 toll alternate^ route study. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 475 curve. The superposed single line is the corresponding estimate by EE, theory of the hour-to-hour call losses. As may be seen, theory and ob- servation are in good agreement both point by point and on the average over the range of losses from 0.01 to 0.50. The dashed line shows the prediction of final route loss for each hour on the assumption that the offered traffic A' was random. Such an assumption gives consistently low estimates of the existing true loss. As of interest, a series of heavy dots is included on Fig. 31. These are the result of calculating the Poisson Summation, P{C,L), where L is the average load carried on, rather than offered to, the C trunks. It is inter- esting that just as in earlier studies in this paper on straight groups of intertoll trunks (for example as seen on Fig. 7), the Poisson Summation with load carried taken as the load offered parameter, gives loss values surprisingly close to those observed. Also, as before, this summation has a tendency to give too-great losses at light loadings of a group and too- small losses at the heavier loadings. ; 7.4 Prediction of Traffic Passing Through a Midti-Stage Alternate Route Network I In the contemplated American automatic toll switching plan, wide I advantage is expected to be taken of the efficiency gains available in i multi-alternate routing. Thus any procedure for traffic analysis and prediction needs to be adaptable for the . more complex multi-stage arrangements as well as the simpler single-stage ones so far examined. Extension of the Equivalent Random theory to successive overflows is easily done since the characterizing parameters, average and variance, of the load overflowing a group of paths are ahvays available. Since few cases of more than single-stage automatic alternate routing are yet in operation in the American toll plant, it is not readily possible to check an extension of the theoiy with actual field data. Moreover col- lecting and analyzing observations on a large operating multi-alternate route system would be a comparatively formidable experiment. However, in New York city's local interoffice trunking there is a very considerable development of multi-alternate routing made possible by the flexibility of the marker arrangements in the No. 1 crossbar switching system. None of these overflow arrangements has been observed as a whole, simultaneously and in detail. The Murray Hill-2 data in OST groups reviewed in Section 7.3.2 were among the partial studies which have been made. In connection with studies made just prior to World War II on these 476 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table VII — Sum of Direct Group Overflow Loads, Offered to OST's Average. Variance Theory 86.06 129.5 Observed 87.12 127.4 local multi -alternate route systems, a throwdown was made in 1941 on a proposed trunk plan for the Murray Hill-6 office. The arrangement of : trunks is shown on Fig. 32. Three successive alternate routes, Office Selector Tandems (OST), Crossbar Tandem (XBT), and Suburban: Tandem (ST), are available to the large majority of the 123 direct trunk groups leading outward to 169 distant offices. (The remaining 46 parcels of traffic did not have direct trunks to distant offices but, as indicated on the diagram, offered their loads directly to a tandem group.) A total of 726 trunks is involved, carrying 475 erlangs of traffic. A throwdown of 34,001 offered calls corresponding to 2.7 hours of traffic was run. Calls had approximate exponential holding times, averag- ing 135 seconds. Records were kept of numbers of calls and the load from the traffic parcels offered to each direct group, as they were carried or passed beyond the groups of paths to which they had access. Loads car- ried by each trunk in the system were also observed by means of a 36- second "switch-count." (The results on the 17 OST groups reported in Section 7.3.1 were part of this study.) Comparisons of observation and theory which are of interest include the combined loads to and overflowing the 17 OST's. Observed versus calculated parameters (starting with theory from the original direct group submitted loads) are given in Table VII. The agreement is seen to be very good. The corresponding comparison of total load from all the OST's is given in Table VIII. Again the agreement is highly satisfactory. Not all of the overflow from the OST's was offered to the 22 crossbar tandem trunks; for economic reasons certain parcels by-passed XBT andf were sent directly to Suburban Tandem.* This posed the problem of breaking off certain portions of the overflow from the OST's, to be added"' again to the overflow from XBT. An estimate was needed of the contri bution made by each parcel of direct group traffic to any OST's over flow. These were taken as proportional to the loads offered the OST by each direct group (this assumes that each parcel suffers the same over- * In the toll alternate route system by -passing of this sort will not occur. Tt U'lnnt uunu u L \'^\\\\\\\'' \va p^^^^^ \\nii\^ S m T^ m: Tr± 7 J '''.': ; , ±±± ^PP tf+Ff- 4^ :« tn ^ '/V//'/-'//./'/''///;^ : fl NO. 16 NO. 17 SPECIAL TANDEM tt inttttiiittttMitiftt ttinit tit t tttttttttt ttttt (V OJ ^O ^^»*^u-^^OsO r- u-^Ty fv^ O OJ fVOJ O^ rNvO TO-* -t OJ C^ V\vO (V-*fV<HrH OI^C^ »H -tTOTO _*(^-*«) -*CJtO <-ITOnO C*- -* ^O r^ O -* J- -*rj r^ -j-joj rj rH O OO 0«> C^OiJ^<*\<*\fH TO^OC^OJU^r^ ^ -^ i/n f^ OtO to r^*rfc"~'-*r^r^rsi CT^^OJi-tiH OO -* r^ i-H ^ O rH ^ rH.-H .-H rH rH rHrH O O O OO O O O rH (-1 ^ O O O O r^rH O O OOOOOOOOOO OOOOO 27.46 5.81 5.44 0.31 5.99 1.96 tOvO -JvO rH fV f'^0^_JO>JD O^ r^rH (Vr- ifNvO »A(NJryNO -*fV (NiTO Of^Of^O^ -* ^ ^ -* (N* ONf^ "^tO C^ t*^C^ <^ *'^ C^CM^^TO >-i t£) E-« cr: tn CO ,_j ti3 < <i<i: tH w kJ »Jo- J i-H w hJ> a. i*; M <ow»H'J<-< mo:-* > M<fHO-<>JM*s<:w mmooo crossbar office. FINAL ,0 TANDEM TRUNKS 5 INTERMEDIATE TANDEM TRUNKS 10 FIRST ALTERNATE 5 ■ nUTE(OST) TRUNKS SUBURBAN TANDEM N0.1 N0.2 N0.3 N0.4 N0.5 N0.6 NO. 7 N0.8 N0.9 NO.IO N0.11 N0.12 N0.I3 N0.14 DIRECT INTEROFFICE TRUNKS 15 r. 10rzE 5:E: 1 --- m ill ]M \M\ Vw timii ttimlmtt tiitifti! ti flmi tttitttttiiit tttitttft tmmmntti tiiittt tmiitittittit tmtmitmitmtni imttt tii i ititnttit ftitt Y I k.ni.L' IMnu ■-! fv IV iH r>j >0 J t^ i-t^.H p^tov>r^s>r^ O (v rt o (N to mr^pj O cvj ^ -tto ''^-4 O rsi tT>C'tJ't~-t^C^NO&''">-i -* (m lAi-i r\ O O- r~ f^ 'O [ERLAN6S) f|JOOO£>r-t^O 'OcvtJ. rHC'O'OO-Ov i^-0~0,0'0 J ■/■OO Ot^fv-^Cr'ovrH O<0 t^i-^f^r^rufOcjpHu^ pJc-j^^^^^OcJ (> 68.91 37.49 60.62 38.49 12.51 48.62 57.42 DESTINATION *Cr^-*t^r>^,^ r.^>r OFFICES S£S253S£ gS? ~-»f«Ot- Jr" mff-tO toO^iT ^rHrHO rH p- ^ -i f-i rt -J r4 -^ ,H ^d O -4 rH r4 O O O O O O «/^ (V cJ .-* O O O -- r+^ f"* O OO O (VrHodcHO ^ddOrHOOOOOOOOOO 12.96 16.96 9.52 16.43 6.88 21.42 8.05 11.97 27.46 5.81 5.44 0.31 5.99 1. T Q Ita. tfc< C3 O Z OW < f) Fig. 32. — Multi-alternate route trunking arrangcinenl at Murray Hill — 6 (New York) local No. 1 crossbar office. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 477 flow probability). The variance of this overflow portion by-passing XBT was estimated by assigning to it the same variance-to-average ratio as was found for the total load overflowing the OST. Subtracting the means and variances so estimated for all items by-passing XBT, left an approxi- mate load for XBT from each OST. Combining these corrected overflows gave mean and variance values for offered load to XBT, Observed values Table VIII - -Sum of Loads Overflowing OST's Theory Observed Avftraere 26.64 58.42 25.92 Variance 61.32 Table IX — Load Offered to Crossbar Tandem I Average. Variance Theory 25.18 47.67 Observed 25.51 56.10 0.10 r -RANDOM TRAFFIC -THROWDOWN ,--NEGATIVE BINOMIAL to 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS P^n THEORY OBSD 1.0 I — - — -^ .^^^ ( ) ( ) ^X AVG 25.18 25.51 0.8 ^V VAR 47.67 56.10 0.6 VS. 0.4 0.2 , , 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- nomial theory versus throwdown observations. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 477 flow probability). The variance of this overflow portion by-passing XBT was estimated by assigning to it the same variance-to-average ratio as was found for the total load overflowing the OST. Subtracting the means ' and variances so estimated for all items by-passing XBT, left an approxi- mate load for XBT from each OST. Combining these corrected overflows gave mean and variance values for offered load to XBT, Observed values Table VIII - - Sum OF Loa Ds Overflowing OST's Theory Observed Average Variance 26.64 58.42 25.92 61.32 Table IX . — Load Offered to Crossbar Tandem Theory Observed Average 25.18 47.67 25.51 Variance 56.10 0.10 r ^-.--RANDOM TRAFFIC -THROWDOWN --NEGATIVE BINOMIAL 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS THEORY OBSD ( ) ( ) AVG 25.18 25.51 VAR 47.67 56.10 Pin 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- nomial theory versus throwdown observations. 478 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table X — Load Overflowing Crossbar Tandem Average. Variance , Observed 6.47 33.48 and those calculated (in the above manner) are given in Table IX. Fig. 33 shows the distribution of XBT offered loads, observed and calcu- lated. The agreement is very satisfactory. The random traffic (Poisson) distribution, is of course, considerably too narrow. In a manner exactly similar to previous cases, the Ecjuivalent Random load method was applied to the XBT group to obtain estimated param- eters of the traffic overflowing. Comparison of observation and theory at this point is given in Table X. Fig. 34 shows the corresponding observed and calculated distributions 0.15 0.10 f(n) 0.05 )RANDOM TRAFFIC THEORY OBSD AVG 6.55 6.47 VAR 23.80 33.48 ^'NEGATIVE BINOMIAL 5 10 15 20 25 30 35 n=NUMBER OF SIMULTANEOUS CALLS P^n _^^RANDOM TRAFFIC --NEGATIVE BINOMIAL THROWDOWN 5 10 15 20 25 30 35 n = NUMBER OF SIMULTANEOUS CALLS Fig. 34 — Distribution of calls from crossbar tandem trunks; negative binomial theory versus throwdown observations. ! THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 479 of siniiiltaneoiis calls. The agreement again is reasonably good, in spite of the considerable disparity in variances. The overflow from XBT and the load which by-passed it, as well as some other miscellaneous parcels of traffic, were now combined for final offer to the Suburban Tandem group of 17 trunks. The comparison of parameters here is again available in Table XI. On Fig. 35 are shown the observed and calculated distributions of simultaneous calls for the load offered to the ST trunks. The agreement is once again seen to be very satisfactory. We now estimate the loss from the ST trunks for comparison with the actual 'proportion of calls which failed to find an idle path, and finally Table XI — Load Offered to Suburban Tandem Average. . Variance . Theory 15.38 42.06 Observed 14.52 48.53 THEORY OBSD f(n) P^n 10 20 30 40 n = NUMBER OF SIMULTANEOUS CALLS I.O ^ ^ \ 0.8 " ^ , --NEGATIVE BINOMIAL 0.6 V ^-THROWDOWN \ \ 0.4 0.2 x^^^ 1 " -r-^ 10 20 30 40 n^NUMBER OF SIMULTANEOUS CALLS 50 Fig. 35 — Distribution of load offered to suburban tandem trunks; negative linomial theory versus throwdown observations. 480 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table XII - — Grade of Service on ST Group Theory Obser- vation Observation Load submitted (erlangs) Load overflowing (er- langs) Proportion load over- flowing 15.38 3.20 0.209 14.52 2.63 0.181 Number of calls sub- mitted 1057 Number of calls over- flowing 200 Proportion of calls over- flowing 0.189 Table XIII — Grade of Service on the System Total load submitted Total load overflowing Proportion of load not served Theory Observed 475 erlangs 3.20 erlangs 0.00674 34,001 calls 200 calls 0.00588 compare the proportions of all traffic offered the system which failed to find a trunk immediately. See Tables XII and XIII. After these several and varied combinations of offered and overflowed loads to a system of one direct and three alternate routes it is seen that 'i the final prediction of amount of load finally lost beyond the ST trunks is gratifyingly close to that actually observed in the throwdown. The prediction of the system grade of service is, of course, correspondingly good. It is interesting in this connection to examine also the proportions I overflowing the ST group when summarized by parcels contributed from the several OST groups. The individual losses are shown on Fig. 36; they appear well in line with the variation one would expect from group to group with the moderate numbers of calls which progressed this far through the multiple. 0.4 0.3 octr o^ 0.2 So a ^0.1 ,-THEORY =0.21 ._>. • • --AVG OBSD = 0.19 12 4 6 8 10 12 14 16 18 20 OST GROUP NUMBER Fig. 36 — Overflow calls on third alternate (ST) route. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 481 7.4.1 Correlation of Loss with Peakedness of Components of Non-Ran- dom Offered Traffic Common sense suggests that if several non-random parcels of traffic are combined, and their joint proportion of overflow from a trunk group is P, the parcels which contain the more peaked traffic should experience overflow proportions larger than P, and the smoother traffic an overflow proportion smaller than P. It is by no means clear however, a priori, the extent to which this would occur. One might conjecture that if any one parcel's contribution to the total combined load is small, its loss would be caused principally by the aggregate of calls from the other parcels, and consequently its own loss would be at about the general average loss P, and hence not very much determined by its own peakedness. The Murray Hill-6 throwdowai results may be examined in this respect. The mean and variance of each OST-parcel of traffic, for example, arriving at the final ST route was recorded, together with, as noted before, its own proportion of overflow from the ST trunks. The variance/mean over- dispersion ratio, used as a measure of peakedness, is plotted for each parcel of traffic against its proportion of loss on Fig. 37. There is an un- doubted, but only moderate, increase in proportion of overflow with increased peakedness in the offered loads. It is quite possible, however, that by recognizing the differences be- tween the service given various parcels of traffic, significant savings in final route trunks can be effected for certain combinations of loads and trunking arrangements. Of particular interest is the service given to a parcel of random traffic offered directly to the final route when compared 04 o oc_l 0.3 UJUJ >u °% °ia2|- zo o< OO0.1 o a. • • • 0.5 1.0 1.5 2.0 2.5 3.0 3.5 V/a OF EACH OST PARCEL REACHING ST TRUNKS Fig. 37 — Effect of peakedness on overflow of a parcel of traffic reaching an ilternate route. 482 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 with that received by non-random parcels overflowing to it from high usage groups. 7.5 Expected Loss on First Routed Traffic Offered to Final Route The congestion experienced by the first-routed traffic offered to the final group in a complex alternate route arrangement [such as the right hand parcels in Figs. 10(c) and (d)] \vill be the same as encountered in a series of random tests of the final route by an independent observer, that is, it will be the proportion of time that all of the final trunks are busy. As noted before, the distribution of simultaneous calls n (and hence the congestion) on the C final trunks produced by some specific arrange- ment of offered load and high usage trunks can be closely simulated by that due to a single Equivalent Random load offered to a straight group of aS -f C trunks. Then the proportion of time that the C trunks are busy in such an equivalent system provides an estimate of the corres- ponding time in the real system ; and this proportion should be approxi- mately the desired grade of service given the first routed traffic. Brockmeyer has given an expression (his equation 36) for the pro- portion of time, Rx , in a simple S -\- C system with random offer A, and "lost calls cleared," that all C trunks are busy, independent of the condition of the *S-trunks: R, = f{S,C,A) = Ii,x,s+cKA) — — where m=o \ m / (S — m However, (rdS) is usually calculated more readily step-by-step using the formula <Tc{S) = aciS - 1) -f CTc-liS) , starting with crc(O) = 1 and ao(S) = A^Sl The average load carried on the C paths is clearly Lc = A[Ei,sU) - Ei,s+c{A)], (31) THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 483 and the variance of the carried load can be shown to be* Vc = ALc ^ - ACEx,s+c{A) + Lc- L\ (32) On Fig. 38, Ri values are shown in solid line curves for several com- binations of A and C over a small range of S trunks. The corresponding losses Ri for all traffic offered the final group, where R^ = oc'/A', are shown as broken curves on the same figure. The R2 values are always above Ri , agreeing with the common sense conclusion that a random component of traffic will receive better service than more peaked non- random components. However, there are evidently considerable areas where the loss differ- ence between the two Z^'s will not be large. In the loss range of principal interest, 0.01 to 0.10, there is less proportionate difference between the R's, as the A = C paired values increase on Fig. 38. For example, at /?2 = 0.05, and A = C = 10, R./Ri = 0.050/0.034 = 1.47; while for A = C = 30, i?2/Ri = 0.050/0.044 = 1.13. Similarly for A = 2C, the R2/R1 ratios are given in Table XIV. Again the rapid decrease in the R2/R1 ratio is notable as A and C increase. F. I. Tange of the Swedish Telephone Administration has performed elaborate simulation studies on a variety of semi-symmetrical alternate route arrangements, to test the disparity between the Ri and R2 types of losses on the final route. f For example if g high-usage groups of 8 paths each, jointly overflow 2.0 erlangs to a final route which also serves 2.0 erlangs of first routed traffic, Tange found the differences in losses between the two 2-erlang parcels, i?high usage (h.u.) —Ri, shown in column 9 of Table XV. The corresponding ER calculations are performed in columns 2 to 8, the last of which is comparable with the throwdown \alues of column 9. The agreement is not unreasonable considering the sensitiveness of determining the difference between two small prob- abilities of loss. A quite similar agreement was found for a variety of other loads and trunk arrangements. * In terms of the first two factorial moments of n : Vc is given by Vc = M(2) + M(i) - M(i)*, where Mw = Lc (leneral expressions Mu) for the factorial moments of n are derived in an unpub- lished memorandum by J. Riordan. t Optimal Use of Both-Way Circuits in Cases of Unlimited Availability, a paper by F. I. T&nge, presented at the First International Congress on the Appli- cation of the Theory of Probability in Telephone Engineering and Administration, June 1955, Copenhagen. 484 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 D O a. _( 0.1 < z 0.09 0.08 II z o 0.07 0.06 U1 in 0.05 o _i 0.04 II o 7- 0.03 O (- a. o 0.02 n o a a. ru a. 0.01 a 0.009 z 0.008 < 0.007 cr 0.006 0.005 0.004 0.003 0.002 0.001 ^•>_^ ^^ ;-.^ V '*^^>v •^ 'N^ Y^x. X \ V vv \ \ \ \ \ \ \ \ A = 30 C = 15 s \s\^ \ .' ^ \\v \ n, -^ ^ f- \ >■ \ \ \ \ "S^ \ V \ \ \ \ \ [a \ ""') \ \ \ ^ \X' ^"' o\ \ \ \ \ \ \ A?T iO \ \ \ \ \ R, \'C-. \ \ I A = 20^ rc = io^^ V \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ ' \^ \ \ \ \ \ A \ \ V A = 10,^ C = io \ \ \ \ \ 10 15 20 25 S = *equivalent"number of paths 30 35 Fig. 38 — Comparison of Ri and R2 losses under various load and trunk con- ditions. Table XIV— The R2/R1 Ratios for A = 2C A C Ri/Ri when R2 = 0.05 10 20 30 5 10 15 10.6 3.25 2.44 THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 485 Table XV — Comparison of E.R. Theory and Throwdowns on Disparity of Loss Between High Usage Overflow and Random Offer to a Final Group (8 trunks in each high usage group; 9 final trunks serving 2.0 erlangs high usage overflow and 2.0 erlangs first routed traffic.) Number of Groups of 8 High Usage Trunks ER Theory {A' = 4.0) Tange V A 5 R2=a7A' i?i Rh.u, ~ 2R-L- Ri Rh.u.—Rl= 2{R2 - Ry) Throwdown Rh.u. - Ri (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 5.77 7.51 4.17 0.0375 0.0251 0.0499 0.0248 0.0180 2 5.80 7.50 4.25 0.0383 0.0255 0.0511 0.0256 0.0247 3 5.74 7.44 4.08 0.0369 0.0248 0.0490 0.0242 0.0286 4 5.68 7.30 3.91 0.0362 0.0247 0.0477 0.0230 0.0276 5 5.64 7.20 3.80 0.0355 0.0242 0.0468 0.0226 0.0245 6 5.58 7.06 3.64 0.0350 0.0240 0.0460 0.0220 0.0221 7 5.55 7.00 3.56 0.0345 0.0238 0.0452 0.0204 0.0202 8 5.51 6.91 3.45 0.0335 0.0236 0.0434 0.0198 0.0188 9 5.47 6.81 3.34 0.0325 0.0231 0.0419 0.0188 0.0177 10 5.45 6.76 3.29 0.0312 0.0225 0.0399 0.0174 0.0166 Limited data are available showing the disparity of Ri and Ro in actual operation in a range of load and trunk values well beyond those for which Ri values have been calculated. Special peg count and over- flow registers were installed for a time on the final route during the 1955 Newark alternate route tests. These gave separate readings for the calls from high usage groups, and for the first routed Newark to Pittsburgh calls. Comparative losses for 17 hours of operation over a wide range of loadings are shown on Fig. 39. The numbers at each pair of points give the per cent of final route offered traffic which was first routed (random). In general, approximately equal amounts of the two types of traffic were offered. In 6 of the hours almost identical loss ratios were observed, in 7 hours the overflow-from-high-usage calls showed higher losses, and in 4 hours lower losses, than the corresponding first routed calls. The non-random calls clearly enjoyed practically as good service as the random calls. This result is not in disagreement with what one might expect from theory. To compare directly with the Newark-Pittsburgh case we should need curves on Fig. 38 expanded to correspond to A', V values of (50, 85) to (120, 200). Examining the mid-range case of C = 65, A' = 70, V = 120, we find A = 123, >S = 54. Here A is approximately 2C; extrapolat- ing the A = 2C curves of Fig. 38 to these much higher values of A and C suggests that R2/R1 w^ould be but little different from unity. 486 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 It is clear from the above theory, throwdowns, and actual observa- tion that there are certain areas where the service differences given first routed and high usage trunk overflow parcels of traffic are significant. In Section 8, where practical engineering methods are discussed, curves are presented which permit recognition of this fact in the determination of final trunk requirements. 7.6 Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped Alternate Route In the engineering of alternate route systems it is necessary to deter- mine the point at which to limit a high usage group of trunks and send the overflow traffic via an alternate route. This is an economic problem whose solution requires an estimate of the load which will be carried on 1.0 0.5 z o il' 0.05 a. UJ > o z o I- cc o a. O a. a- O.OiO 0.005 0.00)0 6 64 56 8' 57 61( OL 65^ 69, 40 56 o 50 ,58 41 58 69 8 64 52 s 38 6 66 49 8 52 O FIRST ROUTED TRAFFIC (NUMBERS INDICATE PER CENT OF TOTAL WHICH IS FIRST ROUTED) • OVERFLOW TRAFFIC FROM 7 HIGH USAGE GROUPS 60 70 80 90 100 110 120 A'= ESTIMATED OFFERED LOAD TO PITTSBURGH IN ERLANGS (INCLUDING RETRIALS) Fig. 39 — Comparison of losses on final route (Newark to Pittsburgh) for high usage overflow and first routed traffic. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 487 the last trunk of a straight high usage group of any specified size, carry- ing either first or higher choice traffic or a mixture thereof.* The Equivalent Random theory readily supplies estimates of the loads carried by any trunk in an alternate routing network. After having found the Equivalent Random load A offered to *S + C trunks which corresponds to the given parameters of the traffic offered to the C trunks, it is a simple matter to calculate the expected load i on any one of the C trunks if they are not slipped or reversed. The load on the ith trunk in a simple straight multiple (or the S + jth. in a divided multiple of *S lower and C upper trunks), is A- = Is+j = A[E^,s+j-M) - Ex,s+j{A)] (33) where Ei,n(A) is the Erlang loss formula. A moderate range of values of ■Ci versus load A is given on Figure 40. f Using this method, selected comparisons of theoretical versus observed loads carried on particular trunks at various points in the Murray- Hill-6 throwdown are shown in Fig. 41 ; these include the loads on each of the trunks of the first two OST groups of Fig. 32, and on the second and third alternate routes, crossbar and suburban tandem, respectively. The agreement is seen to be fairly good, although at the tail end of the latter two groups the observed values drop aw^ay somewhat from the theoretical ones. There seems no explanation for this beyond the possi- bility that the throwdown load samples here are becoming small and might by chance have deviated this far from the true values (or the arbitrary breakdown of OST overflows into parcels offered to and by- passing XBT may well have introduced errors of sufficient amount to account for this disparity). As is well known, (33) gives good estimates of the loads carried by each trunk in a high usage group to which random (Poisson) traffic is offered; this relationship has long been used for the purpose in Bell System trunk engineering. 8. PRACTICAL METHODS FOR ALTERNATE ROUTE ENGINEERING To reduce to practical use the theory so far presented for analysis of alternate route systems, working curves are needed incorporating the * The proper selection point will be where the circuit annual charge per erlang of traffic carried on the last trunk, is just equal to the annual charge per erlang of traffic carried by the longer (usually) alternate route enlarged to handle the overflow traffic. t A comprehensive table of /< is given by A. Jensen as Table IV in his book "Moe's Principle," Copenhagen, 1950; coverage is for / ^ 0.001 erlang, z = 1(1)140; A = 0.1(0.1)10, 10(1)50, 50(4)100. Note that n + 1, in Jensen's notation, equals i here. 488 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 6 <0 6 ID 6 6 6 soNvibB Ni viNnai Hi-n 3hi no agiaavD avon = '-Tf THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 489 pertinent load-loss relationships. The methods so far discussed, and proposed for use, will be briefly reviewed. The dimensioning of each high usage group of trunks is expected to be performed in the manner currently in use, as described in Section 7.6. The critical figure in this method is the load carried on the last high usage trunk, and is chosen so as to yield an economic division of the offered load between high usage and alternate route trunks. Fig. 40 is one form of load-on-each-trunk presentation suitable for choosing eco- nomic high usage group size once the permitted load on the last trunk is established. The character (average a and variance v) of the traffic overflowing each high usage group is easily found from Figs. 12 and 13 (or equivalent - OST GROUP NO. 2 1.U OST GROUP NO.l 0.5 - ^ 4 5 6 TRUNK NUMBER CROSSBAR TANDEM GROUP Z O 2 4 6 8 10 12 14 16 18 20 22 TRUNK NUMBER SUBURBAN TANDEM GROUP 12 4 6 8 10 12 14 16 TRUNK NUMBER Fig. 41 — Comparison of load carried by each alternate route trunk; theory versus throwdowns. 490 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 tables). The respective sums of the overflow a's and v^s, give A' and V by (28) and (29); they provide the necessary statistical description of traffic offered to the alternate route. According to the Equivalent Random method for estimating the alter- nate route trunks required to provide a specified grade of service to the overflow traffic A', one next determines a random load A which when submitted to S trunks will yield an overflow with the same character {A', V) as that derived from the complex system's high usage groups. An alternate route of C trunks beyond these S trunks is then imagined. The erlang overflow a', with random offer A, to S + C trunks is found from standard i^i-formula tables or curves (such as Fig. 12). The ratio R2 = a! I A' is a first estimate of the grade of service given to each parcel of traffic offered to the alternate route. As discussed in Sec- tion 7.5, this service estimate, under certain conditions of load and trunk arrangement, may be significantly pessimistic when applied to a first routed parcel of traffic offered directly to the alternate route. An improved estimate of the overflow probability for such first routed traffic was found to be R\ as given by (30). 8,1 Determination of Final Group Size with First Routed Traffic Offered Directly to the Final Group When first routed traffic is offered directly to the final group, its service Ri will nearly always be poorer than the overall service given to those other traffic parcels enjoying high usage groups. The first routed traffic's service will then be controlling in determining the final group size. Since Ri is a function of *S, C and A in the Equivalent Random solution (30), and there is a one-to-one correspondence of pairs of A and S values with A' and V values, engineering charts can be constructed at selected service levels Ri which shoAv the final route trunks C required, for any given values of A' and V. Figs. 42 to 45 show this relation at service levels of Ri = 0.01, 0.03, 0.05 and 0.10, respectively.* * On Fig. 42 (and also Figs. 46-49) the low numbered curves assume, atjfirst sight, surprising shapes, indicating that a load with given average and variance would require fewer trunks if the average were increased. This arises from the sensitivity of the tails of the distribution of offered calls, to the V'/A' peaked- ness ratio which, of course, decreases with increases in A'. For example, with C = 4 trunks and fixed V = 0.52, the loss rapidly decreases with increasing A': A' V'/A' A S a' a' /A' 0.28 0.33 0.40 0.52 1.86 1.58 1.30 1.00 6.1 3.0 1.42 0.52 10. 5.0 2.03 0.0155 0.0081 0.0036 0.0008 0.055 0.025 0.009 0.002 THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 491 These four Ri levels would appear to cover the most used engineering range. For example, if the traffic offered to the final route (including the first routed traffic) has parameters A' = 12 and V ^ 20, reading on Fig. 43 indicates that to give P = 0.03 "lost calls cleared" service to the first routed traffic, C = 19 final route trunks should be provided. (For random traffic (F' = A' = 12), 17.8 trunks would be required.) Other charts, of course, might be constructed from which Ri could be read for specific values of A', V and C. They would become voluminous, however, if a wide range of all three variables were required. 8.2 Provision of Trunks Individual to First Routed Traffic to Equalize Service If the difference between the service Ri given the first routed parcel of traffic and the service given all of the other parcels, is material, it may be desirable to take measures to diminish these inequities. This may readily be accomplished by setting aside a number of the otherwise full access final route trunks, for exclusive and first choice use of the first routed traffic. High usage groups are now provided for all parcels of traffic. The alternate route then services their combined overflow. The overall grade of service given the ith. parcel of offered traffic in a single stage alter- nate route system will then be approximately '* Pi = Ei,Xi{ai)R2 = EiXiia.)^, (34) Thus the service will tend to be uniform among the offered parcels when all send substantially identical proportions of their offered loads to the alternate route. And the natural provision of "individual" trunks for the exclusive use of the first routed traffic would be such that the same pro- portion should overflow as occurs in the associated high \isage groups. This procedure cannot be followed literally since high usage group size is fixed b}^ economic considerations rather than any predetermined overflow value. The resultant overflow proportions will commonly vary over a considerable range. In this circumstance it would appear reason- able to estimate the objective overflow proportion to be used in estab- lishing the individual group for the first routed traffic, as some weighted average h of the overflow proportions of the several high usage groups. Thus with weights g and overflow proportions h, h = ^'^' + ^'^' + • ' • (35) ^1 + ^2+ • • • * Although not exact, this equation can probably be accepted for most engi- neering purposes where high usage trunks are provided for each parcel of traffic. \ \ •* \ / \ O-l SO o N ■-^ / \ . V— k -^ / \ a — ' ^^ GO eg . r— ^ "< \ b <M ^ fc J _ ^, ^ ^ >\ ;;;;::; "^ . — — r"— b' q to o 6 si - ■ -— [ _^ — — ' \_^^ 1 rj \ >. s; imp- 0<V ^ k^ >^ S \ O/V \ ■*> ^ ^ \ 2\/ K en "t o (\J 7 < 1 (M Ct (\J UJ Z o (\j 1- -) C) 00 tr 1 < ID z or Al Lll 11 11 o O Q < () _l on UJ O < (r LU > < II ^ < tn It t\J o to (C ^f r\j O n ro n ri (M (\J OJ OJ i\J ainoH nvNid Oi aaaaddo avoi dO 30NviavA=,A - \ y\ 0.5 1.0 \ \ \ . \ \ \ "V A \ X ^___ ,^ ^ \ ^ ^ ><: -^ ^ \ A. ^ OJ s^ --- "^ ^ -\ c 3 d O s OJ ^ sS I^ -^ h. OJ >c .—- - ^ -;:: --^ b^ '""^o ^^.; -N. ^0*-, ^ ^v,^ s ^ — ^ \ ^ . .'-^ >X — VL ^ '< Xj ^ D o o cr < z fe '■ DO < o _1 UJ o < q: <0 UJ > < II <o ■^ (\J o <o <o ■<t ry o ri n n n (M <\j <\j (\) (M 3inoa ivNid oi aaaaddo avcn do aoNviMVA=,A 492 ' 1 \ U1 1 § \ . \ <*) Q - in d \ s \, "X^^ \ - ^ \ c \ . ^ \ m ■v. \ ? \ ^ ^ \~ ;; \ X \ ^^ ^ \ 1. ! A ; \ Ok ^^^ -^ ^^ CO , jX. in o >n c ^\^ i nv ' r~- <c -\ k 1 ^ in ^ ^ 1 j \^ 1 '\r K> ' ^ 1 f\i' "* ;. =^-^ — k J ■^A, 1 2" ■^^ O/v Oi ^ t^ 1 (OV^ \ L._ . ' ^f\10<D<0^f\JO(0<0^f>JO<0(£i^OJO rrionrvjruryrvirvj — — — ^ — ainoa ivNid oi asaaddo avoi dO 30NvidVA=,A \ \ q - in d o - \ \ \. ^ \ i r ' ^ ^ \ ' . N ^ ^ ! ^1 >^ . . \ \ ^ ( J ^^'^s,. — - , X U1 O "1 o o rvj 2 N ' — ' ^s 2 <>1 S^ — \ ^% >==:: — ■ ^ i 04., ^ —^ OA/ V; a^Vfc. ^s ^ <. \ ^ k 1 1 1 1 1 ^ J c > a t C 3i 5 r t J r VNId U C 'J 0± Q: 5 a J )HBdd( D u D OVO 4 3 •< n do 93 i 1 30NV U c avA = -A D U 5 ■< X ^ J o Tf <n z < rvj LU i: Z c3 3 O m ki ml > bO O CO a 3 -1^ (\J q: LU t-i U- , , o LL o < C to o e^-i _l o 0} UJ r-. o o • < •t;o 10 LU .2-^ > >o < o ■f II 1 i^ rvj "5 o O bD « (O o Cli E (O <\j -T3 0) ^ If) o 4J /. o < » OJ _l ■4-3 ft tc LU !-. 7 «3 o OJ on LU > 1- -5 00 g O -u _| to < ^ (D ^i C Li- ? P -fj ■<t K <u U -tJ nj Ct lU o Li- u. ,_^ o o3 o n a < n «a _) o cu LU LO < o ^ Ct .-.IC <o LU .^O i, >o II y II Ph c^ ^ Ct-H '^ O . n> u CJ • fH t^H > tH 0) <u 494 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 A choice of all weights g equal to unity will often be satisfactory for the present purpose. The desired high usage group size for the first routed traffic is then found from standard £'i-tables showing trunks x required, as a function of offered traffic a and proportion overflow b. Since the different parcels of traffic have varying proportions h of their ' loads overflowing to the final route, by equation (34) the parcel with the largest proportion will determine the permitted value of R2 . Thus ' R2 = P/&max (36) where P is the specified poorest overall service (say 0.03) for any parcel. It may be noted that on occasion some one parcel, perhaps a small one, may provide an outstandingly large bmax value, which will tend to give a considerably better than required service to all the major traffic parcels. Some compromise with a literal application of a fixed poorest service criterion may be indicated in such cases. An alternative and somewhat simpler procedure here is to use an average value b in (36) instead of ^max , with a compensating modifica- , tion of F, so that substantially the same R2 is obtained as before. The allowance in P will be influenced by the choice of weights g in (35). It will commonly be found in practice that overflow proportions to final groups for large parcels of traffic are lower than for small parcels. Choos- ing all weights, as unity, as opposed to weighting by traffic volumes for : example, tends to insert a small element of service protection for those , traffic parcels (often the smaller ones) with the higher prportionate high . usage group overflows. Having determined R2 , a ready means is needed for estimating the required number of final route trunks. Curves for this purpose are pro- vided on Figs. 46 to 49, within whose range, R2 = 0.01 to 0.10, it will usually be sufficiently accurate to interpolate for trunk engineering purposes. These F2-curves exactly parallel the i?i-curves for use when first routed traffic is offered directly to the final group without benefit of individual high usage trunks. If R2 is well, outside the charted range a run-through of the ER calculations may be required. 8.3 Area in Which Significant Savings in Final Route Trunks are Realized by Allowing for the Preferred Service Given a First Routed Traffic Parcel Considerable effort has been expended by alternate route research workers in various countries to discover and evaluate those areas where first routed (random) traffic ofl'ered to a final route enjoj^s a substantial service advantage over competing parcels of traffic which have over- , THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 495 flowed from high usage groups. A comparison of Figs. 42 to 45, (which indicate trunk provision for meeting a first routed traffic criterion Ri) with Figs. 46 to 49 (which indicate trunk provision for meeting a com- posite-load-offered-to-the-final-route criterion R2) gives a means for de- ciding under what conditions in practice it is important to distinguish between the two criteria. Fig. 50 shows the borders of areas, defined in terms of A' and V, the characterizing parameters of the total load offered to the final route, where a 2 and 5 per cent overprovision of final trunks would occur using R2 for Ri as the loss measure for first routed traffic. Thus in the alternate route examples displayed in Table XV, where x = S, g = 2 to 10, A' = 4.0 and V varies from 5.80 to 5.45, Fig. 50 shows that by failing to allow for the preferred position of the 2 erlang first routed parcel, we should at R = 0.02 engineered loss, provide a little over 5 per cent more final trunks than necessary. (Actually 10.2 and 9.9 versus 9.6 and 9.4 trunks f or gr = 2 and 10; respectively.) The curves of Fig. 50 for final route loads larger than a few erlangs, are almost straight lines. At an objective engineering base of i? = 0.03, for example, the 2 and 5 per cent trunk overprovision areas through using i?2 instead of Ri are outlined closely by: 2 per cent overprovision occurs at Fy(A' — 1) = 1.4 5 per cent overprovision occurs at V'/(A' — 1) = 1.8. Thus in the range of loads covered by Fig. 50, one might conclude that useful and determinable savings in final trunks can be achieved by use of the specialized /?i-curves instead of the more general 7?2-curves, when the ratio V'/(A' — 1) exceeds some figure in the 1.4 to 1.8 range, say 1.6. (In the examples just cited the V'/{A' — 1) ratio is approximately 1.9.) 8.4. Character of Traffic Carried on Non-Final Routes Telephone traffic which is carried by a non-final route will ordinarily be subjected to a peak clipping process which will depress the variance of the carried portion below that of the offered load. If this traffic ter- minates at the distant end of the route, its character, while conceivably affecting the toll and local switching trains in that office, will not require further consideration for intertoll trunk engineering. If, however, some or all of the route's load is to be carried on toll facilities to a more distant point (the common situation), the character of such parcels of traffic will l)e of interest in providing suitable subsequent paths. For this purpose it will be desirable to have etimates of the mean and variance of these carried parcels. When a random traffic of "a" erlangs is offered to a group of "c" paths \ \ q 6 o \ \. ^ ^ ^N \ \ [^ ^ .^^ ^ ^ \ \ o CM ,— • ^__„„— - .^ ^ ^ ^ ^ \\r\j ^ -- — -^ \ 1 H ^ to >< ^ ^ *\ »n o lo c <?, K ^^ ^ ^ x> ^ ^ , ^ ^ ^ \ '^'^. x] ^^ ^ X ^vj y / \ ^ ^ \ ^ ^*nJ ^ to k a> (v «> (M •f (d (\J o 7 < f\l _J f\l ir ?■ O fVJ ILI t- -) en C) ir 1 < <o z Q UJ Q < 3 UJ < a: « UJ (0 < (M O <0 <o ^ (\J O <») 10 M fO fVJ <\l (M (M (M ainob "ivNid oi a3a3ddo a\»cn jo 3DNviavA=,A \ 1 \ ) 0.5 1.0 1 1 1 1 1 1 \ \ \ \ \ \ \ ^ ^ x ^ ^ X ^ ^ ^ ^ ^ ^ ^ ^ ^ rn (\j ^ CM ^Ni ^ ^ ^^ ^ ^ ri q vO c d OJ rvj >c ^ ^ \ V -;;; ^-^ ^^ ^ '"Vo ^^^. H,° ">r ^ /-^ ^ X. ^ 7 ^^^^. '0 r / / U/v ^ l^ V > V i^ ^^ k OO OJ lO C\J <* </l M 0> ?• < (\J _1 a UJ 7 o (\I UJ t- "3 OO <1 a. \ (0 < z u. 2 P n Ul (\1 cr UJ u. u. CJ O n < o -J OO UJ O < fr (O UJ > 4. K> ■* fVJ o CO <o •<r AJ o n («) n rn <\J f\J (\j (\j (\J Binoa ivNid 01 aaaaddo avon do 3DNviavA=,A 496 < . K r -' ^ N 1 O 6 o - ■"^ \. K ^ (J < - _-— -= \ \l \ -= -- \ "^ \ \ rvj uj O ^ . —■ \ ^^^ ^ ^ K . — ' -^ ^^ -- \ ^s^ ^ "" _l < (0 z CO ^ ^ , ■ "^ \= q o c ^ LL c < c---' ^ \ \ ILI (\J >u s-i, ^ ^ ^" - u. u. O O Q '-^'^q ^'^. i^K "^ ^ , N _J Oa, ^ s< y It ^ ^ >s II r^ ^ is •c' 4^ ^ ^ r r r VI c r 3 r VJ r ■< y r i a D il s ^ f r y c 3 a 3 U a ^ f N o ainoa nvNid ojl oaaaddo avoi do 30NviyvA=,A ■^ rvi o CO <o ^ (NJ o fO ro m rvj (\J rvj (\j (M 3inoa nvNid oi aaaajdo avon do 30NvibVA= a 497 498 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 2 4 6 8 10 12 14 16 18 20 22 24 26 A' == AVERAGE LOAD OFFERED TO FINAL ROUTE IN ERLANGS 28 Fig. 50 — Overprovision of final route trunks when R2 is used instead of Ri as service to first routed traffic. and overflowing calls do not return, the variance of the carried load is Fed = a[l - Er, , (a)] fl + aE^, , (a) - aEi, , _i(a)]* (37) and the ratio of variance to average of the carried load is V cd = 1 - a [£-1,0-1 (a) - Ei,c(a)]* = 1 - /c (38) These particular forms are due to P. J. Burke. THEORIES FOR TOLL TRAFFIC EXGINEERIXG IX THE U. S. A. 499 From (38) it is easy to see that Fed = L{1 - Q = (Load carried by the group) (1 — load on last trunk) (39) This is a convenient relationship since for high usage trunk study work, both the loads carried (in eriangs) on the group and on the last trunk will ordinarily be at hand. If the high usage group's load is to be split in various directions at the distant point for re-offer to other groups, it would appear not un- reasonable to assign a variance to each portion so as to maintain the ratio expressed in eciuation (38). That is, if a carried load L is divided into parts Xi , X2 • • • where L = Xi -f X2 • • • , then the associated variances 71 , 72 . . • would be 71 = Xi (1 - fc) y, = Xo (1 - fc) (40) If, however, the load offered to the group is non-random (e.g., the group is an intermediate route in a multi-alternate route system), the procedure is not quite so simple as in the random case just discussed. Equation (32) expresses the variance Vc of the carried load on a group of C paths whose 'offered traffic consists of the overflow from a first group of S paths to which a random load of A eriangs has been offered. Vc could of course be expressed in terms of A', V and C, and curves or tables constructed for working purposes. However, such are not avail- able, and in any case might be unwieldy for practical use. A simple alternative procedure can be used which jdelds a conserva- tive (too large) estimate of carried load variance. With random load offered to a divided two stage multiple of x paths followed by tj paths, a positive correlation exists between the numbers m and n of calls present simultaneously on the x and y paths, respectively. Then the variance V-n+n of the m -\- n distribution is greater than the sum of the individual variances of m and n, y m-\-n ^ ' m l~ ' n or Vm < y^n - Vn (41) Now n can be chosen arbitrarily, and if made very large, Vm+n becomes the offered load variance, and F„ the overflow load variance. Both of these are usually (or can be made) available. Their difference then, according to (41) gives an upper limit to F,„ , the desired carried load 500 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table XVI — Approximate Determination of the Variance OF Carried Loads; X lower paths, 8 upper paths; offer to upper paths = 3 erlangs Lower Paths, x Upper Paths, y No. Lower Paths X Random offered load A (= V) Variance of overflow Vn Estimated variance of carried load V -Vn True variance of carried load Eq (37) Variance of offer V (= Vn) (Col 3) Variance of overflow V" Estimated variance of cd load V - V" True variance of cd load (Brocli- meyer) (1) 3 6 12 (2) 3.00 5.399 7.856 12.882 (3) 3.00 4.05 4.98 6.22 (4) 1.35 2.88 6.66 (5) 0.60 1.418 3.538 (6) 3.00 4.05 4.95 6.22 (7) 0.035 0.121 0.236 0.520 (8) 2.97 3.93 4.74 5.70 (9) 2.853 3.664 4.175 4.790 variance- Corresponding reasoning yields the same conclusion when the offered load before the x paths is non-random. A numerical example by Brockmeyer" while clearly insufficient iu establish the degree of the inequality (41), indicates something as to the discrepancy introduced by this approximate procedure. Comparison with the true values is shown in Table XVI. In the case of random offer to the 0, 3, 6, 12 "lower paths," the ap- proximate method of equation (41) overestimates the variance of the carried load by nearly two to one (columns 4 and 5 of Table XVI). The exact procedure of (37) is then clearly desirable when it is applicable, that is when random traffic is being offered. For the 8 upper paths to which non-random load is offered (the non-randomness is suggested by comparing the variance of column 6 in Table XVI with the average offered load of 3 erlangs), the approximate formula (41) gives a not too extravagant overestimate of the true carried load variance. Until curves or tables are computed from equation (32), it would appear useful to follow the above procedure for estimating the carried load variance when non-random load is offered. 8.5. Solution of a Typical Toll Multi- Alternate Route TrunJcing Arrange- ment: Bloomsburg, Pa. In Fig. 9 a typical, moderately complex, toll alternate route layout was illustrated. It is centered on the toll office at Bloomsburg, Pa. The loads to be carried between Bloomsburg and the ten surrounding cities are indicated in CCS (hundred call seconds per hour of traffic; 36 CCS = 1 erlang). The numbers of direct high usage trunks shown are assumed to have been determined by an economic study; we are asked to find I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 501 the number of trunks which should be installed on the Bloomsburg- Harrisburg route, so that the last trunk will carry approximately 18 CCS (0.50 erlang). Following this determination, (a) the number of final trunks from Bloomsburg to Scranton is desired so that the poorest service given to any of the original parcels of traffic will be no more than 3 calls in 100 meeting NC. Also (6) the modified Bloomsburg-Scranton trunk arrangement is to be determined when a high usage group is pro- vided for the first routed traffic. Solution (a): First Routed Traffic Offered Directly to Final Group The offered loads in CCS to each distant point are shown in column (2) of Table XVII; the corresponding erlang values are in column (3). Consulting Figs. 12 and 13, the direct group overflow load parameters, average and variance, are read and entered in columns (5) and (6) re- spectively for the four groups overflowing to Harrisburg, and in columns (7) and (8) for the four groups directly overflowing to Scranton. The variance for the direct Bloomsburg-Harrisburg traffic equals its average ; likewise for the direct Bloomsburg-Scranton traffic. They are so entered in the table. The parameters of the total load on the Harrisburg group are found by totalhng, giving A' = 11.19, and V = 19.90. The required size Ci of the Harrisburg group is now determined by the Equivalent Random theory. Entering Fig. 25 with A' and V just determined, the ER values of trunks and load found are Si = 13.55, and Ai = 23.75. Ci is to be selected so that on a straight group of Si + Ci trunks with offered load A, the last trunk will carry 0.50 erlang. Reading from Fig. 40, the load carried by the 26th trunk approximates this figure. Hence Ci = 26 — *Si = 12.45 trunks; or choose 12 trunks. The overflow load's mean and variance from the Harrisburg group v/ith 12 trunks, is now read from Figs. 12 and 13, entering with load Ai = 23.75 and Ci -\- Si = 25.55 trunks. The overflow values (a' = 2.50 and v' = 7.50) are entered in columns (7) and (8) of the table. The total offered load to Scranton is now obtained by totalling columns (7) and (8), giving A" = 16.27 and V" = 25.60. We desire now to know the number of trunks C2 for the Scranton group which will provide NC 3 per cent of the time to the poorest service parcel of traffic, i.e., the first routed Bloomsburg-Scranton parcel. The Ri = 0.03 and R2 = 0.03 solutions are available, the former of course being more closely applicable. A check reference to Fig. 50 shows a difference of approximately 4 per cent in trunk provision would result from the two methods. Entering Figs. 43 and 47 with A" = 16.27 and 502 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 J ¥ >- O 0) o 05 l-H CO ^ lO • CCI O --I CO CO "O CO lO I-H >* ,X ^ lO ^ (M <wIt-H Oi-H (M (N -IJ aj P I- o '-' 03 ■* CO U5 <M I-H 'a • pH S-i P^ (N --1 OOOO g X oddd d I-H d II § "o "o o l-o 3 cS II m m o o |2 < o O o o I-H O 00 t^ '^ o ( -* 00 C3 CO >o I-H CO m o -4-* o dcO I-H I-H d T-H J OO "3 -tj pq o a l> II £3 03 % 3 o ^ O. 02 O OOOcO -* t^ M •na iO(M OOOT I-H (M CO Iz; 5g di-n'dd d d o o 2^ c4 t^ s I-H T-H II H c<> 05 P - ^ (M ic ic o CO t^ o II o o rrj coco o t^ dec t^i-H 05 T-H > H §§• P> be O li ti H ►J o E kC *< ^ CO bO ^ co' r-H « II W tn J^ O OCDCO t^ Oi Eh •n.ss cot^co o ^ T-H ^ CM O <m" (M .-H -<J< l-H ^ 2K 43 S bO in I-H 11 bD a o u ^ CO ^ ^ o "S^^ ■* < bi) J C S' s ^ rf T-HIO OiC ^ ICGOOOIO P .-H(M o (M a ■ OJ C S o ;h «4H < T3 -a w (A oPh t> bO bO (N Oi^ t-- (MC^COOO '^ lO ^— ' H-t L- C ,,-^ l>-0^'* rt< "^i <N CO C^ I-H :3 -2 •^ • co'^' l-H ""^ oiod^" -* COCOCO Tf< d I-H . c "J w .—1 i-H (M I-H II IM H oO ^^^ TJ OQ P <u -^ a> tJ !U (0 M l-H _ CO O r-f O T-H CO CO 00^ >o • rH n fS (M '^ OCO CO (M CO <M lO CO u lO O T-H I-H rH 00(M I-H CO <a l-H t3 til " o > o to XI :3 •♦J < Eh CJ •4-* a ■*-» to 5 - ■4J 1- o (U c ft > ° s s bfi »-. 3 -Q u "H.S •C o 1 ^ o o 11^ o3 P^WJ^Ph THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 503 I Y" = 25.60, we obtain the trunk requirements: ! Rx Method 23.8 trunks i?2 Method 24.8 trunks Thus the more precise method of sokition here yields a reduction of 1 .0 in 25 trunks, a saving of 4 per cent, as had been predicted. The above calculation is on a Lost Calls Cleared basis. Since the over- flow direct traffic calls will return to this group to obtain service, to as- sure their receiving no more than 3 per cent 'NC, the provision of the final route would theoretically need to be slightly more liberal. An esti- mate of the allowance required here may be made by adding the ex- pected erlangs loss A for the direct traffic (most of the final route over- flow calls which come from high usage routes will be carried by their respective groups on the next retrial) to both the A" and Y" values previously obtained, and recalculating the trunks required from that point onward. (In fact this could have been included in the initial com- putation.) Thus: A = 0.03 X 10.14 = 0.30 erlang A'" = 16.27 + 0.30 = 16.57 erlangs V" = 25.60 + 0.30 = 25.90 erlangs Again consulting Figs. 43 and 47 gives the corresponding final trunk values Ri Method 24.1 trunks R2 Method 25.1 trunks Of the above four figures for the number of trunks in the Scranton route, the i?i-Method with retrials, i.e., 24.1 trunks, would appear to give the best estimate of the required trunks to give 0.03 service to the poorest service parcel. Solution (h) : With High Usage Group Provided for First Routed Traffic Following the procedure outlined in Section 8.2, we obtain an average of the proportions overflowing to the final route for all offered load par- cels. The individual parcel overflow proportion estimates are shown in the last column of Table XVII; their unweighted average is 0.112. With a first routed offer to Scranton of 10.14 erlangs, a provision of 12 high usage trunks will result in an overflow of a = 1.26 erlangs, or a propor- tion, of 0.125 which is the value most closely attainable to the objective 0.112. With 12 trunks the overflow variance is found to be 2.80. 504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Replacing 10.14 in columns 7 and 8 of Table XVII with 1.26 and 2.80, respectively, gives new estimates characterizing the offer to the final route. A" = 7.39 and V" — 18.26. We now proceed to insure that the poorest service parcel obtains 0.03 service. This occurs on the Phila- delphia and Harrisburg groups, which overflow to the final group ap- proximately 0.224 of their original offered loads. The final group must then, according to equation (34) be engineered for R2 = 0.03/0.224 = 0.134 service. This value lies above the highest R2 engineering chart (Fig. 49, R2 = 0.10), so an ER calculation is indicated. The Equivalent Random average is 28.6 erlangs, and S = 23.5 trunks. We determine the total trunks S -\- R which, with 28.6 erlangs offered, will overflow 0.134(7.39) = 0.99 erlang. From Fig. 12.2, 35.6 trunks are required. Then the final route provision should be C = 35.6 — 23.5 = 12.1 trunks; and a total of 12 + 12.1 or 24.1 Scranton trunks is indicated. Simplified Alternative Solution: In Section 8.2 a simplified approxi- mate procedure was described using a modified probability P' for the average overall service for all parcels of traffic, instead of P for the poor- est service parcel. Suppose P' = 0.01 is chosen as being acceptable. Then P' 01 «' = T = am = oo^" Interpolating between the R2 = 0.05 and 0.10 curves (Figs. 48 and 49) gives with A" = 7.39 and F" = 18.26, C = 13.4, the number of final trunks required. Again the same result could have been obtained by making the suitable ER computation. It may be noted that if P' had been chosen as 0.015 (one-half of P), R2 would have become 0.134, exactly the same value found in the poorest-service-parcel method. The final trunk provision, of course, would have again l)een 12.1 trunks. Disscussion In the first solution above, 24.1 full access final trunks from Blooms- burg to Scranton were refiuired. The service on the first routed traffic was 0.03; however, the service enjoyed by the offered traffic as a whole was markedly better than 0.03. The corresponding ER calculation shows (.4 = 28.3, .S -\- C = 12.3 + 24.1) a total overflow of a" = 0.72 erlangs, or an overall service of 0.72/91.21 = 0.008. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 505 In the second solution, 12 high usage and 12.1 common final, or a total of 24.1, trunks were again required, to give 0.03 service to the poorest service parcels of offered load. The overall service here, however, was 0.99/91.21 .= 0.011. Thus, with the same number of paths provided, in the second solution (high usage arrangement) the overall call loss was 40 pes cent larger than in the first solution,* However, it may well be desirable to accept such an average service penalty since by providing high usage trunks for the first routed traffic, the latter's service cannot be degraded nearly so readily should heavy overloads occur momentarily in the other parcels of traffic. 9. CONCLUSION As direct distance dialing increases, it will be necessary to provide intertoll paths so that substantially no-delay service is given at all times. To do this economically, automatic multi-alternate routing will replace the present single route operation. Traffic engineering of these compli- cated trunking arrangements will be more difficult than with simple intertoll groups. One of the new problems is to describe adequately the non-random character of overflow traffic. In the present paper this is proposed to be done by employing both mean and variance values to describe each par- cel of traffic, instead of only the mean as used heretofore. Numerous comparisons are made with simulation results which indicate that ade- quate predictive reliability is obtained by this method for most traffic engineering and administrative purposes. Working curves are provided by which trunking arrangements of considerable complexity can readily j be solved. A second problem requiring further review is the day-to-day variation i among the primary loads and their effect on the alternate route system's I grade of service. A thorough study of these variations will permit a re- I evaluation of the service criteria which have tentatively been adopted. j A closely allied problem is that of providing the necessary kind and [ amounts of traffic measuring devices at suitable points in the toll alter- ! nate route systems. Requisite to the solution of both of these problems ! is an understanding of traffic flow character in a complex overflow-type I * The actual loss difference may be slightly greater than estimated here since i in the first solution (complete access final trunks), an allowance was included for i j return attempts to the final route by first routed calls meeting an 0.03 loss, while 1 in the second solution (high usage group for first routed traffic) no return at- i| tempts to the final route were considered. These would presumably be small since I I only 1 per cent of all calls would overflow and most of these upon retrial would be ij handled on their respective high usage groups. 506 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of trunking plan, and a method for estimating quantitatively the essential fluctuation parameters at each point in such a system. The present paper has undertaken to shed some light on the former, and to provide an approximate j^et sufficiently accurate method by which the latter can be accomplished. It may be expected then that these studies, as they are developed, will provide the basis for assuring an adequate direct dis- tance dialing service at all times with a minimum investment in intertoll trunk facilities. ACKNOWLEDGEMENTS The author wishes to acknowledge the technical and mathematical as- sistance of his associates, Mrs. Sallie P. Mead, P. J. Burke, W. J. Hall, and W. S. Hayward, in the preparation of this paper. Dr. Hall provided the material on the convolution of negative binomials leading to Fig. 19. Mr. Hayward extended Kosten's curve E on Fig. 5 to higher losses by a calculating method involving the progressive squaring of a probability matrix. The author's thanks are also due J. Riordan who has summarized | some of the earlier mathematical work of H. Nyquist and E. C. INIolina, as well as his own, in the study of overflow load characteristics; this appears as Appendix I. The extensive calculations and chart constructions are principally the work of Miss C. A. Lennon. REFERENCES 1. Rappleye, S. C, A Study of the Delays Encountered bj'^ Toll Operators in Ob- taining an Idle Trunk, B. S.T.J. , 25, p. 539, Oct., 1946. 2. Kosten, L., Over de Invloed van Herhaalde Oproepen in de Theorie der Blok- keringskausen, De Ingenieur, 59, j). 1'j123, Nov. 21, 1947. 3. Clos, C, An Aspect of the Dialing Behavior of Subscribers and Its Effect on the Trunk Plant, B. S.T.J. , 27, p. 424, July, 1948. 4. Kosten, L., Uber Sperrungswahrscheinlichkeiten bei Staffelschaltungen, E.N.T., 14, p. 5, Jan., 1937. 5. Kosten, L., Over Blokkeerings-en Wachti)rol>lemen, Thesis, Delft, 1942. 6. Molina, E. C, Appendix to: Interconnection of Telephone Systems — Graded Multiples (R. I. Wilkinson), B.S.T.J., 10, p. 531, Oct., 1931. 7. Vaulot, A. E., Application du Calcul des Probabilites a I'Exploitation Tele- phonique. Revue Gen. de I'Electricite, 16, p. 411, Sept. 13, 1924. 8. Lundcpiist, K., General Theorv for Telephone Traffic, Ericsson Technics, 9, p. Ill, 1953. 9. Berkeley, G. S., Traffic and Trunking Principles in Automatic Telei)hony, 2nd revised edition, 1949, Ernest Benn, Ltd., London, Chapter V. 10. Pahu, C., Calcul I']xact de la Perte dans les Groupes de Circuits Echelonn^s, lOricsson Technics, 3, ]). 41, 1936. 11. Brockmever, 1']., The Simph> Overflow Problem in the Theory of Telephone Traffic! Teleteknik, 5, ji. 361, December, 1954. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 507 ABRIDGED BIBLIOGRAPHY OF ARTICLES ON TOLL ALTERNATE ROUTING Clark, A. B., and Osborne, H. S., Automatic Switching for Nationwide Telephone Service, A.I.E.E., Trans., 71, Part I, p. 245, 1952. (Also B.S.T.J., 31, p. 823, Sept., 1952.) Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, A.I.E.E. Trans., 71, Part I, p. 248, 1952. (Also B.S.T.J., 31, p. 832, Sept., 1952.) Nunn, W. H., Nationwide Numbering Plan, A.I.E.E. Trans., 71, Part I, p. 257, 1952. (Also B.S.T.J., 31, p. 851, Sept., 1952.) Clark, A. B., The Development of Telephony in the United States, A.I.E.E. Trans., 71, Part I, p. 348, 1952. Shiplev, F. F., Automatic Toll Switching Systems, A.I.E.E. Trans., 71, Part I, p. '261, 1952. (Also B.S.T.J., 31, p. 860, Sept., 1952.) Myers, O., The 4A Crossbar Toll System for Nationwide Dialing, Bell Lab. Record, 31, p. 369, Oct., 1953. Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Lab. Record, 32, p. 51, Feb., 1954. Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- ments in Alternate Routing Trunk Networks, B.S.T.J., 33, p. 277, March, 1954. Molnar, I., Some Recent Advances in the Economy of Routing Calls in Nation- wide Dialing, A.E. Tech. Jl., 4, p. 1, Dec, 1954. Jacobitti, E., Automatic Alternate Routing in the 4A Crossbar System, Bell Lab. Record, 33, p. 141, April, 1955. Appendix I* DERIVATION OF MOMENTS OF OVERFLOW TRAFFIC This appendix gives a derivation of certain factorial moments of the c(iuilibrium probabilities of congestion in a di^dded full-access multiple used as a basis for the calculations in the text. These moments were de- rived independently in unpublished memoranda (1941) by E. C. Molina (the first four) and by H. Nyquist; curiously, the method of derivation here, which uses factorial moment generating functions, employs auxili- ary relations from both Molina and Nyquist. Although these factorial moments may be obtained at a glance from the probability expressions given by Kosten in 1937, if it is remembered that pw = |:(-i)'-'(';)^, (1.1) where p{x) is a discrete probability and M (k) is the A;th factorial moment of its distribution, Kosten does not so identify the moments and it may 1)0 interesting to have a direct derivation. Starting from the equilibrium formulas of the text for f(;ni, n), the l)robability of m trunks busy in the specific group of x trunks, and n in Prepared by J. Riordan. 508 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 the (unlimited) common group, namely {a -{- m -\- n)f(m, n) — (w + l)f(m + 1, n) — (n + l)/(m, n + 1) — af(in — 1, n) = (1-2) « (a -{- X -{- n)j{x, n) — af{x, n — 1) \ - (n -\- l)f(x, n + 1) - af(x - 1, n) = and /(m, n) = 0, m < or n < or m > x, factorial moment generating function recurrences may be found and solved. With m fixed, factorial moments of n are defined by M(fc)(m) = E {n)kf{m, n) (1.3) n=0 or alternatively by the factorial moment exponential generating function M{m, = Z MUm)t'/k\ = £ (1 + 07K n) (1.4) ] fc=0 n=0 I In (1.3), {n)k = n{n — 1) • • • (n — /c + 1) is the usual notation for a \ falling factorial. Using (1.4) in equations (1.2), and for brevity D = d/dt, it is found that a^ m ^- tD)M{in, t) - (m + l)M{m + 1, t) - aM(m - l,t) = (1.5) (x - at -\- tD)M{x, t) - aM{x - \,t) = which correspond (by equating powers of t) to the factorial moment re- currences {a-\- m^ k)M^kM) - (m + l)Ma)(w + 1) - ailf (fc)(m - 1) = (1.6) (x + k)M(k)(x) - akM^k-i)ix) - aMik)(x - 1) = Notice that the first of (1.6) is a recurrence in m, which suggests (fol- lowing Molina) introducing a new generating function defined by Gdu) = T.M^k){m)u'^ (1.7) THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 509 Using this in (1.5), it is found that (a -h k - au + (u - l)~\ GM = (1.8) Hence 1 dGM^^^J^ ^j_g^ Gk(u) du I — u and, by easy integrations, Gk{u) = ce"" (1 - ur\ (1.10) with c an arbitrary constant, which is clearly identical with Gk(0) = M(.)(0). Expansion of the right-hand side of (1.10) shows that il/a,(m) = Ma)(0) Z "^ •^. , "" ■„ = Ma,(0)a-.(m), (1.11) j=o \ J / {m - j)l if <jo{m) = a'/ml and, a,(w) = ^ ( •^- ~ ) y-^ ^ri (1-12) The notation ak(m) is copied from Xyquist; the functions are closely related to the ^^^"^ used by Kosten; indeed akim) = e'ipm'''' ■ They have the generating function 00 Qkiu) = 53 (TkMu" = e""(l — u)~'' (1.13) from which a number of recurrences are found readily. Thus Qkiu) = (1 - u)gk+Xu) u -^ — = augkiu) + kugk+i(u) du = -agk-iiu) + (a - k)gk(u) + kgk-i(u) (the last by use of the first) imply ckim) = ak+iim) — (Tk+iim — 1) m(Tk(m) = ackim — 1) + k<jk+i(m — 1) = - a<jk-i{m) + (a - k)<Tk{m) + k<Xk+i(ni) 510 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 The first of these leads to cr/^(0) + (7,(1) + • • • + cr,(x) = ak+i(x) (1.14) J and the last is useful in the form j kak+i{m) = {m + k - a)ak{m) + 0(rt_i(m) (1.15) ■- Also, the first along with ao(m) = a" /m\ leads to a simple calculation ; procedure, as Kosten has noticed. By (1.11) the factorial moments are now completely determined ex- cept for il/(A-)(0). To determine the latter, the second of (1.6) and the normalizing equation X E M,{m) = 1 (1.16) are available. Thus from the second of (1 .6) [(:r + k)<r,{x) - mu{x - l)].^/(A-)(0) = a/v(r,_i(.c)M(,_i)(0) (1.17) Also {x + k)ak{x) — acTkix — 1) = (x -\- k - a)ak(x) + a[(Xk{x) - (Tk(.x — 1)] = (x -\- k - a)(Tk{x) + a<Tk-i{x) = /t'o-fc+i^r), the last step by (1.15). Hence (Tk-l{x) MaM = a "-^=^ Ma-iM (1.18) <rk+i{x) and by iteration ^k (7i(x)(roix) MaAO) = a' "^7" "7, Mo(0) (1.19) From (l.ll) and (1.16), and in the last step (1.14), t.M,(m) = i: il/o(0)cro(7n) = ilfo(0)cri(a;) = 1 (1.20) Hence finally Ma)(m) = Ma-M<rk{ni) , a,(x)ak(m) (1.21) = a (Tki.i{.r)<^k{x) THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 511 ^""^ Ma) = Z Ma,{m) = a'ao{x)/a,(x) (1.22) m=0 Ordinary moments are found from the factorial moments by linear relations; thus if Wt is the A;th ordinary moment (about the origin) mo = M^o) nil = M^) m-i = il/(2) + il/(i) mz = il/(3) + 3Af (2) + il/(i) Thus mo(m) = (ro(m)/ai(x) mi(m) = aai(m)(To(x) / (Ti(x)(T2(x) vi-iim) = aa2{m)(TQ{x) / (r'2{x)(7z{x) + a(Ji{;m)<Ta{x) / <ti{x)(T2{x) and, in particular, using notation of the text mo{x) = (ro{x)/ax{x) = Ei,xia) mi(x) (Tiix) a (Xx = — ^r = a mo{x) <T.(x) .T - a + 1 + aEi,,{a) (1.23) ni2{x) 2 aaiix) , 2 Vx = — 7-r — (Xx = ir-^ + OCx — ax , ^ mo{x) csix) (1,24) = ax[l — ax + 2a(x + 2 -\- ax - a)~^] X Finally the sum moments: nik = ^ mk{m) are Wo = 1 mi = a = a(To{x)/(yi{x) = aEi_x{a) rrh = aaQ{x)/a2{x) -\- mi = mi[a{x -\- I -\- nii — a)~ +1] (1.25) (1.26) y = m2 — mi = mi[l — vh + a(.^' + 1 + nii — a) ] In these, Ei,x(a) = (ro(:c)/(ri(.T) is the familiar Erlang loss function. Appendix II — character of overflow load when non-random TRAFFIC IS offered TO A GROUP OF TRUNKS It has long been recognized that it would be useful to have a method by which the character of the overflow traffic could be determined when non-random traffic is offered to a group of trunks. Excellent agreement has been found in both throwdown and field observation over ranges of considerable interest with the "equivalent random" method of describ- to z < LJJ in z D h- X o en Q < o 3 LL cr LU > o LL o LLI < LJJ II 0.04f/ AQ-- 0.02 TRUN 10 .1 0:3 TRUNKS .1 0.3 1.0 3 a